A different way of thinking of numbers [duplicate]

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP












27















This question already has an answer here:



  • Mapping natural numbers into prime-exponents space

    4 answers



I was more or less day dreaming about whole numbers, and I had an idea which seemed novel to me, but which may in fact be previously known and/or studied. I'm not even sure what to call it, or how to look for previous reports of it. My first question will be: Have I merely rediscovered a way of looking at things that is already known, and if so, where can I find some discussion of it?



I imagine an infinite dimension space, with each orthogonal axis corresponding to a prime number. By the fundamental theorem of arithmetic, natural numbers each have a unique representation of the form $prod p_i^alpha_i$. Numbers of this form can be located in $n$-dimensional subspaces as follows: For each prime $p_j$ which has a corresponding exponent $>0$, move $alpha_1$ units from the origin along the $p_1$ axis, then $alpha_2$ units parallel to the $p_2$ axis, then $alpha_3$ units parallel to the $p_3$ axis, etc., for as many primes as are present as factors of that number. The origin would correspond to the number $1$, where all axes intersect and $alpha_i=0$ for all $i$.



Each axis could be extended through the origin, and fractions could be represented as negative integral distances along the axes so extended, allowing the representation of rational numbers. Interestingly, moving "infinitely" out along these negative axes, one approaches $0$.



Next, by moving non-integral distances along axes, non-integral exponents could be represented and algebraic rational numbers could be represented and located in appropriate subspaces. I'm not certain that transcendental numbers can be represented, but neither are they representable in the form $prod p_i^alpha_i$. One concern I have is that if one is allowed to move irrational distances along axes, it might conceivably be possible to represent a particular number in two different ways, and locate it in two different subspaces, i.e. if exponents can be identified such that $p_1^alpha_1=p_2^alpha_2$.



I do not think arithmetic addition/subtraction can be readily performed with numbers so represented, but multiplication/division can be. In this light, this kind of representation bears some kinship to logarithms, but with a unique (prime) base (rather than $e$ or $10$) as the metric along each axis. However, I can think of no particular utility to representing numbers in this way. So my second question is: Can anyone see any merit or utility to thinking of numbers in this way?










share|cite|improve this question













marked as duplicate by Bill Dubuque number-theory
Users with the  number-theory badge can single-handedly close number-theory questions as duplicates and reopen them as needed.

StackExchange.ready(function()
if (StackExchange.options.isMobile) return;

$('.dupe-hammer-message-hover:not(.hover-bound)').each(function()
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');

$hover.hover(
function()
$hover.showInfoMessage('',
messageElement: $msg.clone().show(),
transient: false,
position: my: 'bottom left', at: 'top center', offsetTop: -7 ,
dismissable: false,
relativeToBody: true
);
,
function()
StackExchange.helpers.removeMessages();

);
);
);
Dec 21 '18 at 2:52


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.










  • 3




    What a fascinating idea!
    – DonielF
    Dec 20 '18 at 3:57






  • 2




    You might be interested in the Gelfond-Schneider Theorem and the p-adic numbers.
    – Anguepa
    Dec 20 '18 at 3:59






  • 1




    @YiFan: Which group do you mean by "[t]his..group"?
    – user 170039
    Dec 20 '18 at 4:07






  • 2




    You really want to restrict to integral coordinates on those axes. So you’ll be getting a very nice representation of the positive rational numbers. As @Anguepa suggests, you should look into the $p$-adic numbers; and also the idèles and adèles.
    – Lubin
    Dec 20 '18 at 4:08






  • 2




    It's even been considered in physics, albeit with non-negative coefficients/exponents only: en.wikipedia.org/wiki/Primon_gas
    – J.G.
    Dec 20 '18 at 11:02















27















This question already has an answer here:



  • Mapping natural numbers into prime-exponents space

    4 answers



I was more or less day dreaming about whole numbers, and I had an idea which seemed novel to me, but which may in fact be previously known and/or studied. I'm not even sure what to call it, or how to look for previous reports of it. My first question will be: Have I merely rediscovered a way of looking at things that is already known, and if so, where can I find some discussion of it?



I imagine an infinite dimension space, with each orthogonal axis corresponding to a prime number. By the fundamental theorem of arithmetic, natural numbers each have a unique representation of the form $prod p_i^alpha_i$. Numbers of this form can be located in $n$-dimensional subspaces as follows: For each prime $p_j$ which has a corresponding exponent $>0$, move $alpha_1$ units from the origin along the $p_1$ axis, then $alpha_2$ units parallel to the $p_2$ axis, then $alpha_3$ units parallel to the $p_3$ axis, etc., for as many primes as are present as factors of that number. The origin would correspond to the number $1$, where all axes intersect and $alpha_i=0$ for all $i$.



Each axis could be extended through the origin, and fractions could be represented as negative integral distances along the axes so extended, allowing the representation of rational numbers. Interestingly, moving "infinitely" out along these negative axes, one approaches $0$.



Next, by moving non-integral distances along axes, non-integral exponents could be represented and algebraic rational numbers could be represented and located in appropriate subspaces. I'm not certain that transcendental numbers can be represented, but neither are they representable in the form $prod p_i^alpha_i$. One concern I have is that if one is allowed to move irrational distances along axes, it might conceivably be possible to represent a particular number in two different ways, and locate it in two different subspaces, i.e. if exponents can be identified such that $p_1^alpha_1=p_2^alpha_2$.



I do not think arithmetic addition/subtraction can be readily performed with numbers so represented, but multiplication/division can be. In this light, this kind of representation bears some kinship to logarithms, but with a unique (prime) base (rather than $e$ or $10$) as the metric along each axis. However, I can think of no particular utility to representing numbers in this way. So my second question is: Can anyone see any merit or utility to thinking of numbers in this way?










share|cite|improve this question













marked as duplicate by Bill Dubuque number-theory
Users with the  number-theory badge can single-handedly close number-theory questions as duplicates and reopen them as needed.

StackExchange.ready(function()
if (StackExchange.options.isMobile) return;

$('.dupe-hammer-message-hover:not(.hover-bound)').each(function()
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');

$hover.hover(
function()
$hover.showInfoMessage('',
messageElement: $msg.clone().show(),
transient: false,
position: my: 'bottom left', at: 'top center', offsetTop: -7 ,
dismissable: false,
relativeToBody: true
);
,
function()
StackExchange.helpers.removeMessages();

);
);
);
Dec 21 '18 at 2:52


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.










  • 3




    What a fascinating idea!
    – DonielF
    Dec 20 '18 at 3:57






  • 2




    You might be interested in the Gelfond-Schneider Theorem and the p-adic numbers.
    – Anguepa
    Dec 20 '18 at 3:59






  • 1




    @YiFan: Which group do you mean by "[t]his..group"?
    – user 170039
    Dec 20 '18 at 4:07






  • 2




    You really want to restrict to integral coordinates on those axes. So you’ll be getting a very nice representation of the positive rational numbers. As @Anguepa suggests, you should look into the $p$-adic numbers; and also the idèles and adèles.
    – Lubin
    Dec 20 '18 at 4:08






  • 2




    It's even been considered in physics, albeit with non-negative coefficients/exponents only: en.wikipedia.org/wiki/Primon_gas
    – J.G.
    Dec 20 '18 at 11:02













27












27








27


6






This question already has an answer here:



  • Mapping natural numbers into prime-exponents space

    4 answers



I was more or less day dreaming about whole numbers, and I had an idea which seemed novel to me, but which may in fact be previously known and/or studied. I'm not even sure what to call it, or how to look for previous reports of it. My first question will be: Have I merely rediscovered a way of looking at things that is already known, and if so, where can I find some discussion of it?



I imagine an infinite dimension space, with each orthogonal axis corresponding to a prime number. By the fundamental theorem of arithmetic, natural numbers each have a unique representation of the form $prod p_i^alpha_i$. Numbers of this form can be located in $n$-dimensional subspaces as follows: For each prime $p_j$ which has a corresponding exponent $>0$, move $alpha_1$ units from the origin along the $p_1$ axis, then $alpha_2$ units parallel to the $p_2$ axis, then $alpha_3$ units parallel to the $p_3$ axis, etc., for as many primes as are present as factors of that number. The origin would correspond to the number $1$, where all axes intersect and $alpha_i=0$ for all $i$.



Each axis could be extended through the origin, and fractions could be represented as negative integral distances along the axes so extended, allowing the representation of rational numbers. Interestingly, moving "infinitely" out along these negative axes, one approaches $0$.



Next, by moving non-integral distances along axes, non-integral exponents could be represented and algebraic rational numbers could be represented and located in appropriate subspaces. I'm not certain that transcendental numbers can be represented, but neither are they representable in the form $prod p_i^alpha_i$. One concern I have is that if one is allowed to move irrational distances along axes, it might conceivably be possible to represent a particular number in two different ways, and locate it in two different subspaces, i.e. if exponents can be identified such that $p_1^alpha_1=p_2^alpha_2$.



I do not think arithmetic addition/subtraction can be readily performed with numbers so represented, but multiplication/division can be. In this light, this kind of representation bears some kinship to logarithms, but with a unique (prime) base (rather than $e$ or $10$) as the metric along each axis. However, I can think of no particular utility to representing numbers in this way. So my second question is: Can anyone see any merit or utility to thinking of numbers in this way?










share|cite|improve this question














This question already has an answer here:



  • Mapping natural numbers into prime-exponents space

    4 answers



I was more or less day dreaming about whole numbers, and I had an idea which seemed novel to me, but which may in fact be previously known and/or studied. I'm not even sure what to call it, or how to look for previous reports of it. My first question will be: Have I merely rediscovered a way of looking at things that is already known, and if so, where can I find some discussion of it?



I imagine an infinite dimension space, with each orthogonal axis corresponding to a prime number. By the fundamental theorem of arithmetic, natural numbers each have a unique representation of the form $prod p_i^alpha_i$. Numbers of this form can be located in $n$-dimensional subspaces as follows: For each prime $p_j$ which has a corresponding exponent $>0$, move $alpha_1$ units from the origin along the $p_1$ axis, then $alpha_2$ units parallel to the $p_2$ axis, then $alpha_3$ units parallel to the $p_3$ axis, etc., for as many primes as are present as factors of that number. The origin would correspond to the number $1$, where all axes intersect and $alpha_i=0$ for all $i$.



Each axis could be extended through the origin, and fractions could be represented as negative integral distances along the axes so extended, allowing the representation of rational numbers. Interestingly, moving "infinitely" out along these negative axes, one approaches $0$.



Next, by moving non-integral distances along axes, non-integral exponents could be represented and algebraic rational numbers could be represented and located in appropriate subspaces. I'm not certain that transcendental numbers can be represented, but neither are they representable in the form $prod p_i^alpha_i$. One concern I have is that if one is allowed to move irrational distances along axes, it might conceivably be possible to represent a particular number in two different ways, and locate it in two different subspaces, i.e. if exponents can be identified such that $p_1^alpha_1=p_2^alpha_2$.



I do not think arithmetic addition/subtraction can be readily performed with numbers so represented, but multiplication/division can be. In this light, this kind of representation bears some kinship to logarithms, but with a unique (prime) base (rather than $e$ or $10$) as the metric along each axis. However, I can think of no particular utility to representing numbers in this way. So my second question is: Can anyone see any merit or utility to thinking of numbers in this way?





This question already has an answer here:



  • Mapping natural numbers into prime-exponents space

    4 answers







number-theory soft-question






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 20 '18 at 3:49









Keith Backman

9991612




9991612




marked as duplicate by Bill Dubuque number-theory
Users with the  number-theory badge can single-handedly close number-theory questions as duplicates and reopen them as needed.

StackExchange.ready(function()
if (StackExchange.options.isMobile) return;

$('.dupe-hammer-message-hover:not(.hover-bound)').each(function()
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');

$hover.hover(
function()
$hover.showInfoMessage('',
messageElement: $msg.clone().show(),
transient: false,
position: my: 'bottom left', at: 'top center', offsetTop: -7 ,
dismissable: false,
relativeToBody: true
);
,
function()
StackExchange.helpers.removeMessages();

);
);
);
Dec 21 '18 at 2:52


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.






marked as duplicate by Bill Dubuque number-theory
Users with the  number-theory badge can single-handedly close number-theory questions as duplicates and reopen them as needed.

StackExchange.ready(function()
if (StackExchange.options.isMobile) return;

$('.dupe-hammer-message-hover:not(.hover-bound)').each(function()
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');

$hover.hover(
function()
$hover.showInfoMessage('',
messageElement: $msg.clone().show(),
transient: false,
position: my: 'bottom left', at: 'top center', offsetTop: -7 ,
dismissable: false,
relativeToBody: true
);
,
function()
StackExchange.helpers.removeMessages();

);
);
);
Dec 21 '18 at 2:52


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









  • 3




    What a fascinating idea!
    – DonielF
    Dec 20 '18 at 3:57






  • 2




    You might be interested in the Gelfond-Schneider Theorem and the p-adic numbers.
    – Anguepa
    Dec 20 '18 at 3:59






  • 1




    @YiFan: Which group do you mean by "[t]his..group"?
    – user 170039
    Dec 20 '18 at 4:07






  • 2




    You really want to restrict to integral coordinates on those axes. So you’ll be getting a very nice representation of the positive rational numbers. As @Anguepa suggests, you should look into the $p$-adic numbers; and also the idèles and adèles.
    – Lubin
    Dec 20 '18 at 4:08






  • 2




    It's even been considered in physics, albeit with non-negative coefficients/exponents only: en.wikipedia.org/wiki/Primon_gas
    – J.G.
    Dec 20 '18 at 11:02












  • 3




    What a fascinating idea!
    – DonielF
    Dec 20 '18 at 3:57






  • 2




    You might be interested in the Gelfond-Schneider Theorem and the p-adic numbers.
    – Anguepa
    Dec 20 '18 at 3:59






  • 1




    @YiFan: Which group do you mean by "[t]his..group"?
    – user 170039
    Dec 20 '18 at 4:07






  • 2




    You really want to restrict to integral coordinates on those axes. So you’ll be getting a very nice representation of the positive rational numbers. As @Anguepa suggests, you should look into the $p$-adic numbers; and also the idèles and adèles.
    – Lubin
    Dec 20 '18 at 4:08






  • 2




    It's even been considered in physics, albeit with non-negative coefficients/exponents only: en.wikipedia.org/wiki/Primon_gas
    – J.G.
    Dec 20 '18 at 11:02







3




3




What a fascinating idea!
– DonielF
Dec 20 '18 at 3:57




What a fascinating idea!
– DonielF
Dec 20 '18 at 3:57




2




2




You might be interested in the Gelfond-Schneider Theorem and the p-adic numbers.
– Anguepa
Dec 20 '18 at 3:59




You might be interested in the Gelfond-Schneider Theorem and the p-adic numbers.
– Anguepa
Dec 20 '18 at 3:59




1




1




@YiFan: Which group do you mean by "[t]his..group"?
– user 170039
Dec 20 '18 at 4:07




@YiFan: Which group do you mean by "[t]his..group"?
– user 170039
Dec 20 '18 at 4:07




2




2




You really want to restrict to integral coordinates on those axes. So you’ll be getting a very nice representation of the positive rational numbers. As @Anguepa suggests, you should look into the $p$-adic numbers; and also the idèles and adèles.
– Lubin
Dec 20 '18 at 4:08




You really want to restrict to integral coordinates on those axes. So you’ll be getting a very nice representation of the positive rational numbers. As @Anguepa suggests, you should look into the $p$-adic numbers; and also the idèles and adèles.
– Lubin
Dec 20 '18 at 4:08




2




2




It's even been considered in physics, albeit with non-negative coefficients/exponents only: en.wikipedia.org/wiki/Primon_gas
– J.G.
Dec 20 '18 at 11:02




It's even been considered in physics, albeit with non-negative coefficients/exponents only: en.wikipedia.org/wiki/Primon_gas
– J.G.
Dec 20 '18 at 11:02










5 Answers
5






active

oldest

votes


















20














For the natural numbers, this is the same as representing each number by its prime factorization. A number $2^a3^b5^cldots$ is represented by the tuple $(a,b,c,ldots)$ This is often a useful way to represent numbers in number theory, but I don't think the geometric picture adds anything.



Once you allow fractional exponents it is less useful. If you require the exponents be rational you still have a unique representation. For example $sqrt 2=2^1/2$ can only be expressed that way.



If you allow real exponents you no longer have a unique point for each number. You could represent $sqrt 5$ as the point $frac 12$ on the $5$ axis, but you could also represent it as the point $frac 12 log_2(5)$ on the $2$ axis or many other ways.






share|cite|improve this answer


















  • 3




    Would it be a field? I'm not sure it's closed under addition - e.g. can you express $sqrt2+sqrt3$ as a product of rational powers of primes?
    – Carmeister
    Dec 20 '18 at 8:49






  • 1




    @Carmeister You cannot; any number represented by a tuple of rational coefficients is a $k$-th root of a rational number, where $k$ is the gcd of the denominators. That is to say, it is of the form $sqrt[k]q$ for some positive integer $k$ and positive rational number $q$.
    – Servaes
    Dec 20 '18 at 10:34











  • @Carmeister: you are correct. I will remove that.
    – Ross Millikan
    Dec 20 '18 at 14:46










  • if you allow nonrational coordinates, wouldn't that make each number correspond to a (countably) infinite dimensional manifold?
    – WorldSEnder
    Dec 20 '18 at 19:23










  • @WorldSEnder: I believe so.
    – Ross Millikan
    Dec 20 '18 at 19:55


















4














I've found this a useful way to represent the divisors of a number. For example, the divisors of a number like $500$ (whose only prime factors are $2$ and $5$) form a 2D grid:



1 2 4
5 10 20
25 50 100
125 250 500


The divisors of $60$ (which has three prime factors) form a 3D grid, and so on.



This visualization makes many questions about divisors more intuitive to answer. For example, the number of divisors is just the area of the rectangle or cuboid or whichever. The formula for the sum of divisors is easily found by looking at the rows of the rectangle individually. The overlap between the divisors of $400$ and the divisors of $1250$ is the overlap between two rectangles. Some Möbius inversion formulas are just inclusion-exclusion computations on regions of this rectangle.






share|cite|improve this answer
















  • 1




    This connection is discussed in this answer and its comments.
    – Mark S.
    Dec 20 '18 at 18:58


















2














In order for this approach to have any merit, I'd advise you to investigate some properties of this way of working: you are displaying natural numbers as coordinates in a multiple axes system. What about elementary actions, like transformations? (When you transform a number one time into itself, you calculate its square. Are there any properties of transformations you can translate into number theory properties?



What about cryptograpy?



Imagine you have a number, of which you don't know its coordinates. What will be your procedure to find its coordinates? ... Can you find anything in that direction? ...



Good luck






share|cite|improve this answer




























    1














    The idea of representing integers as vectors of their prime factorization exponents is used in some of the fastest factorization algorithms which are based on Dixon's method. This trick allows turning the problem of finding a nontrivial factorization of an integer $n$ into a problem of linear algebra over $mathbbF_2$.



    You first choose a set of prime numbers called the factor base (for example it can be the set of all prime numbers less than a bound $B$). Then you look for lots of numbers $x$ such that $x^2 pmod n$ is $B$-smooth (number that is divisible only by primes less than $B$). You can represent these numbers as vectors in terms of the exponents of the primes occuring in the prime factorization. We only consider the parity (exponent mod $2$) for reasons I'll explain later. The vectors live in a finite-dimensional vector space over $mathbbF_2$ with dimension equal to the size of the factor base.



    Once you have found sufficiently many $B$-smooth numbers ($1$ more than the dimension of the vector space), by general linear algebra you are guaranteed that you have a nontrivial linear dependence among the vectors, and this in turn allows you to produce a nontrivial congruence $x^2 equiv y^2 pmod n$ by combining the relations. The reason we work over $mathbbF_2$ is because we only care about perfect squares, which means that all exponents in the prime factorization should be even.



    With such a congruence of squares, you can use the identity $x^2 - y^2 = (x+y)(x-y)$ and with a high probability, this leads to a nontrivial factorization of $n$ by considering $gcd(n, x + y)$ or $gcd(n, x - y)$ (not guaranteed, but if you do that a couple times it will likely work).



    The fastest general purpose integer factorization algorithm known today is the General Number Field Sieve, and it still uses this idea from Dixon's method fundamentally. The improvements come from more efficient discovery of smooth numbers, by extending the factor bases over rings other than $mathbbZ$.






    share|cite|improve this answer






























      0














      Along with that great data science rule of "graph early", I suggest starting with 2, 3, 5, and 7, i.e. the first four dimensions, and making animated 3D plots showing the allowable coordinates for some example numbers and to see what you get. So:



       2 3 5 7
      first plot (trivial)
      1 0 0 0 0

      second plot
      2 1 0 0 0
      2 0 x 0 0
      2 0 0 y 0
      2 0 z1 z2 0

      third plot
      2.5 . . . .
      2.5 . . . .
      2.5 . . . .





      share|cite|improve this answer



























        5 Answers
        5






        active

        oldest

        votes








        5 Answers
        5






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        20














        For the natural numbers, this is the same as representing each number by its prime factorization. A number $2^a3^b5^cldots$ is represented by the tuple $(a,b,c,ldots)$ This is often a useful way to represent numbers in number theory, but I don't think the geometric picture adds anything.



        Once you allow fractional exponents it is less useful. If you require the exponents be rational you still have a unique representation. For example $sqrt 2=2^1/2$ can only be expressed that way.



        If you allow real exponents you no longer have a unique point for each number. You could represent $sqrt 5$ as the point $frac 12$ on the $5$ axis, but you could also represent it as the point $frac 12 log_2(5)$ on the $2$ axis or many other ways.






        share|cite|improve this answer


















        • 3




          Would it be a field? I'm not sure it's closed under addition - e.g. can you express $sqrt2+sqrt3$ as a product of rational powers of primes?
          – Carmeister
          Dec 20 '18 at 8:49






        • 1




          @Carmeister You cannot; any number represented by a tuple of rational coefficients is a $k$-th root of a rational number, where $k$ is the gcd of the denominators. That is to say, it is of the form $sqrt[k]q$ for some positive integer $k$ and positive rational number $q$.
          – Servaes
          Dec 20 '18 at 10:34











        • @Carmeister: you are correct. I will remove that.
          – Ross Millikan
          Dec 20 '18 at 14:46










        • if you allow nonrational coordinates, wouldn't that make each number correspond to a (countably) infinite dimensional manifold?
          – WorldSEnder
          Dec 20 '18 at 19:23










        • @WorldSEnder: I believe so.
          – Ross Millikan
          Dec 20 '18 at 19:55















        20














        For the natural numbers, this is the same as representing each number by its prime factorization. A number $2^a3^b5^cldots$ is represented by the tuple $(a,b,c,ldots)$ This is often a useful way to represent numbers in number theory, but I don't think the geometric picture adds anything.



        Once you allow fractional exponents it is less useful. If you require the exponents be rational you still have a unique representation. For example $sqrt 2=2^1/2$ can only be expressed that way.



        If you allow real exponents you no longer have a unique point for each number. You could represent $sqrt 5$ as the point $frac 12$ on the $5$ axis, but you could also represent it as the point $frac 12 log_2(5)$ on the $2$ axis or many other ways.






        share|cite|improve this answer


















        • 3




          Would it be a field? I'm not sure it's closed under addition - e.g. can you express $sqrt2+sqrt3$ as a product of rational powers of primes?
          – Carmeister
          Dec 20 '18 at 8:49






        • 1




          @Carmeister You cannot; any number represented by a tuple of rational coefficients is a $k$-th root of a rational number, where $k$ is the gcd of the denominators. That is to say, it is of the form $sqrt[k]q$ for some positive integer $k$ and positive rational number $q$.
          – Servaes
          Dec 20 '18 at 10:34











        • @Carmeister: you are correct. I will remove that.
          – Ross Millikan
          Dec 20 '18 at 14:46










        • if you allow nonrational coordinates, wouldn't that make each number correspond to a (countably) infinite dimensional manifold?
          – WorldSEnder
          Dec 20 '18 at 19:23










        • @WorldSEnder: I believe so.
          – Ross Millikan
          Dec 20 '18 at 19:55













        20












        20








        20






        For the natural numbers, this is the same as representing each number by its prime factorization. A number $2^a3^b5^cldots$ is represented by the tuple $(a,b,c,ldots)$ This is often a useful way to represent numbers in number theory, but I don't think the geometric picture adds anything.



        Once you allow fractional exponents it is less useful. If you require the exponents be rational you still have a unique representation. For example $sqrt 2=2^1/2$ can only be expressed that way.



        If you allow real exponents you no longer have a unique point for each number. You could represent $sqrt 5$ as the point $frac 12$ on the $5$ axis, but you could also represent it as the point $frac 12 log_2(5)$ on the $2$ axis or many other ways.






        share|cite|improve this answer














        For the natural numbers, this is the same as representing each number by its prime factorization. A number $2^a3^b5^cldots$ is represented by the tuple $(a,b,c,ldots)$ This is often a useful way to represent numbers in number theory, but I don't think the geometric picture adds anything.



        Once you allow fractional exponents it is less useful. If you require the exponents be rational you still have a unique representation. For example $sqrt 2=2^1/2$ can only be expressed that way.



        If you allow real exponents you no longer have a unique point for each number. You could represent $sqrt 5$ as the point $frac 12$ on the $5$ axis, but you could also represent it as the point $frac 12 log_2(5)$ on the $2$ axis or many other ways.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 20 '18 at 14:47

























        answered Dec 20 '18 at 4:01









        Ross Millikan

        292k23196371




        292k23196371







        • 3




          Would it be a field? I'm not sure it's closed under addition - e.g. can you express $sqrt2+sqrt3$ as a product of rational powers of primes?
          – Carmeister
          Dec 20 '18 at 8:49






        • 1




          @Carmeister You cannot; any number represented by a tuple of rational coefficients is a $k$-th root of a rational number, where $k$ is the gcd of the denominators. That is to say, it is of the form $sqrt[k]q$ for some positive integer $k$ and positive rational number $q$.
          – Servaes
          Dec 20 '18 at 10:34











        • @Carmeister: you are correct. I will remove that.
          – Ross Millikan
          Dec 20 '18 at 14:46










        • if you allow nonrational coordinates, wouldn't that make each number correspond to a (countably) infinite dimensional manifold?
          – WorldSEnder
          Dec 20 '18 at 19:23










        • @WorldSEnder: I believe so.
          – Ross Millikan
          Dec 20 '18 at 19:55












        • 3




          Would it be a field? I'm not sure it's closed under addition - e.g. can you express $sqrt2+sqrt3$ as a product of rational powers of primes?
          – Carmeister
          Dec 20 '18 at 8:49






        • 1




          @Carmeister You cannot; any number represented by a tuple of rational coefficients is a $k$-th root of a rational number, where $k$ is the gcd of the denominators. That is to say, it is of the form $sqrt[k]q$ for some positive integer $k$ and positive rational number $q$.
          – Servaes
          Dec 20 '18 at 10:34











        • @Carmeister: you are correct. I will remove that.
          – Ross Millikan
          Dec 20 '18 at 14:46










        • if you allow nonrational coordinates, wouldn't that make each number correspond to a (countably) infinite dimensional manifold?
          – WorldSEnder
          Dec 20 '18 at 19:23










        • @WorldSEnder: I believe so.
          – Ross Millikan
          Dec 20 '18 at 19:55







        3




        3




        Would it be a field? I'm not sure it's closed under addition - e.g. can you express $sqrt2+sqrt3$ as a product of rational powers of primes?
        – Carmeister
        Dec 20 '18 at 8:49




        Would it be a field? I'm not sure it's closed under addition - e.g. can you express $sqrt2+sqrt3$ as a product of rational powers of primes?
        – Carmeister
        Dec 20 '18 at 8:49




        1




        1




        @Carmeister You cannot; any number represented by a tuple of rational coefficients is a $k$-th root of a rational number, where $k$ is the gcd of the denominators. That is to say, it is of the form $sqrt[k]q$ for some positive integer $k$ and positive rational number $q$.
        – Servaes
        Dec 20 '18 at 10:34





        @Carmeister You cannot; any number represented by a tuple of rational coefficients is a $k$-th root of a rational number, where $k$ is the gcd of the denominators. That is to say, it is of the form $sqrt[k]q$ for some positive integer $k$ and positive rational number $q$.
        – Servaes
        Dec 20 '18 at 10:34













        @Carmeister: you are correct. I will remove that.
        – Ross Millikan
        Dec 20 '18 at 14:46




        @Carmeister: you are correct. I will remove that.
        – Ross Millikan
        Dec 20 '18 at 14:46












        if you allow nonrational coordinates, wouldn't that make each number correspond to a (countably) infinite dimensional manifold?
        – WorldSEnder
        Dec 20 '18 at 19:23




        if you allow nonrational coordinates, wouldn't that make each number correspond to a (countably) infinite dimensional manifold?
        – WorldSEnder
        Dec 20 '18 at 19:23












        @WorldSEnder: I believe so.
        – Ross Millikan
        Dec 20 '18 at 19:55




        @WorldSEnder: I believe so.
        – Ross Millikan
        Dec 20 '18 at 19:55











        4














        I've found this a useful way to represent the divisors of a number. For example, the divisors of a number like $500$ (whose only prime factors are $2$ and $5$) form a 2D grid:



        1 2 4
        5 10 20
        25 50 100
        125 250 500


        The divisors of $60$ (which has three prime factors) form a 3D grid, and so on.



        This visualization makes many questions about divisors more intuitive to answer. For example, the number of divisors is just the area of the rectangle or cuboid or whichever. The formula for the sum of divisors is easily found by looking at the rows of the rectangle individually. The overlap between the divisors of $400$ and the divisors of $1250$ is the overlap between two rectangles. Some Möbius inversion formulas are just inclusion-exclusion computations on regions of this rectangle.






        share|cite|improve this answer
















        • 1




          This connection is discussed in this answer and its comments.
          – Mark S.
          Dec 20 '18 at 18:58















        4














        I've found this a useful way to represent the divisors of a number. For example, the divisors of a number like $500$ (whose only prime factors are $2$ and $5$) form a 2D grid:



        1 2 4
        5 10 20
        25 50 100
        125 250 500


        The divisors of $60$ (which has three prime factors) form a 3D grid, and so on.



        This visualization makes many questions about divisors more intuitive to answer. For example, the number of divisors is just the area of the rectangle or cuboid or whichever. The formula for the sum of divisors is easily found by looking at the rows of the rectangle individually. The overlap between the divisors of $400$ and the divisors of $1250$ is the overlap between two rectangles. Some Möbius inversion formulas are just inclusion-exclusion computations on regions of this rectangle.






        share|cite|improve this answer
















        • 1




          This connection is discussed in this answer and its comments.
          – Mark S.
          Dec 20 '18 at 18:58













        4












        4








        4






        I've found this a useful way to represent the divisors of a number. For example, the divisors of a number like $500$ (whose only prime factors are $2$ and $5$) form a 2D grid:



        1 2 4
        5 10 20
        25 50 100
        125 250 500


        The divisors of $60$ (which has three prime factors) form a 3D grid, and so on.



        This visualization makes many questions about divisors more intuitive to answer. For example, the number of divisors is just the area of the rectangle or cuboid or whichever. The formula for the sum of divisors is easily found by looking at the rows of the rectangle individually. The overlap between the divisors of $400$ and the divisors of $1250$ is the overlap between two rectangles. Some Möbius inversion formulas are just inclusion-exclusion computations on regions of this rectangle.






        share|cite|improve this answer












        I've found this a useful way to represent the divisors of a number. For example, the divisors of a number like $500$ (whose only prime factors are $2$ and $5$) form a 2D grid:



        1 2 4
        5 10 20
        25 50 100
        125 250 500


        The divisors of $60$ (which has three prime factors) form a 3D grid, and so on.



        This visualization makes many questions about divisors more intuitive to answer. For example, the number of divisors is just the area of the rectangle or cuboid or whichever. The formula for the sum of divisors is easily found by looking at the rows of the rectangle individually. The overlap between the divisors of $400$ and the divisors of $1250$ is the overlap between two rectangles. Some Möbius inversion formulas are just inclusion-exclusion computations on regions of this rectangle.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 20 '18 at 17:19









        Misha Lavrov

        43.9k555104




        43.9k555104







        • 1




          This connection is discussed in this answer and its comments.
          – Mark S.
          Dec 20 '18 at 18:58












        • 1




          This connection is discussed in this answer and its comments.
          – Mark S.
          Dec 20 '18 at 18:58







        1




        1




        This connection is discussed in this answer and its comments.
        – Mark S.
        Dec 20 '18 at 18:58




        This connection is discussed in this answer and its comments.
        – Mark S.
        Dec 20 '18 at 18:58











        2














        In order for this approach to have any merit, I'd advise you to investigate some properties of this way of working: you are displaying natural numbers as coordinates in a multiple axes system. What about elementary actions, like transformations? (When you transform a number one time into itself, you calculate its square. Are there any properties of transformations you can translate into number theory properties?



        What about cryptograpy?



        Imagine you have a number, of which you don't know its coordinates. What will be your procedure to find its coordinates? ... Can you find anything in that direction? ...



        Good luck






        share|cite|improve this answer

























          2














          In order for this approach to have any merit, I'd advise you to investigate some properties of this way of working: you are displaying natural numbers as coordinates in a multiple axes system. What about elementary actions, like transformations? (When you transform a number one time into itself, you calculate its square. Are there any properties of transformations you can translate into number theory properties?



          What about cryptograpy?



          Imagine you have a number, of which you don't know its coordinates. What will be your procedure to find its coordinates? ... Can you find anything in that direction? ...



          Good luck






          share|cite|improve this answer























            2












            2








            2






            In order for this approach to have any merit, I'd advise you to investigate some properties of this way of working: you are displaying natural numbers as coordinates in a multiple axes system. What about elementary actions, like transformations? (When you transform a number one time into itself, you calculate its square. Are there any properties of transformations you can translate into number theory properties?



            What about cryptograpy?



            Imagine you have a number, of which you don't know its coordinates. What will be your procedure to find its coordinates? ... Can you find anything in that direction? ...



            Good luck






            share|cite|improve this answer












            In order for this approach to have any merit, I'd advise you to investigate some properties of this way of working: you are displaying natural numbers as coordinates in a multiple axes system. What about elementary actions, like transformations? (When you transform a number one time into itself, you calculate its square. Are there any properties of transformations you can translate into number theory properties?



            What about cryptograpy?



            Imagine you have a number, of which you don't know its coordinates. What will be your procedure to find its coordinates? ... Can you find anything in that direction? ...



            Good luck







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 20 '18 at 9:43









            Dominique

            31627




            31627





















                1














                The idea of representing integers as vectors of their prime factorization exponents is used in some of the fastest factorization algorithms which are based on Dixon's method. This trick allows turning the problem of finding a nontrivial factorization of an integer $n$ into a problem of linear algebra over $mathbbF_2$.



                You first choose a set of prime numbers called the factor base (for example it can be the set of all prime numbers less than a bound $B$). Then you look for lots of numbers $x$ such that $x^2 pmod n$ is $B$-smooth (number that is divisible only by primes less than $B$). You can represent these numbers as vectors in terms of the exponents of the primes occuring in the prime factorization. We only consider the parity (exponent mod $2$) for reasons I'll explain later. The vectors live in a finite-dimensional vector space over $mathbbF_2$ with dimension equal to the size of the factor base.



                Once you have found sufficiently many $B$-smooth numbers ($1$ more than the dimension of the vector space), by general linear algebra you are guaranteed that you have a nontrivial linear dependence among the vectors, and this in turn allows you to produce a nontrivial congruence $x^2 equiv y^2 pmod n$ by combining the relations. The reason we work over $mathbbF_2$ is because we only care about perfect squares, which means that all exponents in the prime factorization should be even.



                With such a congruence of squares, you can use the identity $x^2 - y^2 = (x+y)(x-y)$ and with a high probability, this leads to a nontrivial factorization of $n$ by considering $gcd(n, x + y)$ or $gcd(n, x - y)$ (not guaranteed, but if you do that a couple times it will likely work).



                The fastest general purpose integer factorization algorithm known today is the General Number Field Sieve, and it still uses this idea from Dixon's method fundamentally. The improvements come from more efficient discovery of smooth numbers, by extending the factor bases over rings other than $mathbbZ$.






                share|cite|improve this answer



























                  1














                  The idea of representing integers as vectors of their prime factorization exponents is used in some of the fastest factorization algorithms which are based on Dixon's method. This trick allows turning the problem of finding a nontrivial factorization of an integer $n$ into a problem of linear algebra over $mathbbF_2$.



                  You first choose a set of prime numbers called the factor base (for example it can be the set of all prime numbers less than a bound $B$). Then you look for lots of numbers $x$ such that $x^2 pmod n$ is $B$-smooth (number that is divisible only by primes less than $B$). You can represent these numbers as vectors in terms of the exponents of the primes occuring in the prime factorization. We only consider the parity (exponent mod $2$) for reasons I'll explain later. The vectors live in a finite-dimensional vector space over $mathbbF_2$ with dimension equal to the size of the factor base.



                  Once you have found sufficiently many $B$-smooth numbers ($1$ more than the dimension of the vector space), by general linear algebra you are guaranteed that you have a nontrivial linear dependence among the vectors, and this in turn allows you to produce a nontrivial congruence $x^2 equiv y^2 pmod n$ by combining the relations. The reason we work over $mathbbF_2$ is because we only care about perfect squares, which means that all exponents in the prime factorization should be even.



                  With such a congruence of squares, you can use the identity $x^2 - y^2 = (x+y)(x-y)$ and with a high probability, this leads to a nontrivial factorization of $n$ by considering $gcd(n, x + y)$ or $gcd(n, x - y)$ (not guaranteed, but if you do that a couple times it will likely work).



                  The fastest general purpose integer factorization algorithm known today is the General Number Field Sieve, and it still uses this idea from Dixon's method fundamentally. The improvements come from more efficient discovery of smooth numbers, by extending the factor bases over rings other than $mathbbZ$.






                  share|cite|improve this answer

























                    1












                    1








                    1






                    The idea of representing integers as vectors of their prime factorization exponents is used in some of the fastest factorization algorithms which are based on Dixon's method. This trick allows turning the problem of finding a nontrivial factorization of an integer $n$ into a problem of linear algebra over $mathbbF_2$.



                    You first choose a set of prime numbers called the factor base (for example it can be the set of all prime numbers less than a bound $B$). Then you look for lots of numbers $x$ such that $x^2 pmod n$ is $B$-smooth (number that is divisible only by primes less than $B$). You can represent these numbers as vectors in terms of the exponents of the primes occuring in the prime factorization. We only consider the parity (exponent mod $2$) for reasons I'll explain later. The vectors live in a finite-dimensional vector space over $mathbbF_2$ with dimension equal to the size of the factor base.



                    Once you have found sufficiently many $B$-smooth numbers ($1$ more than the dimension of the vector space), by general linear algebra you are guaranteed that you have a nontrivial linear dependence among the vectors, and this in turn allows you to produce a nontrivial congruence $x^2 equiv y^2 pmod n$ by combining the relations. The reason we work over $mathbbF_2$ is because we only care about perfect squares, which means that all exponents in the prime factorization should be even.



                    With such a congruence of squares, you can use the identity $x^2 - y^2 = (x+y)(x-y)$ and with a high probability, this leads to a nontrivial factorization of $n$ by considering $gcd(n, x + y)$ or $gcd(n, x - y)$ (not guaranteed, but if you do that a couple times it will likely work).



                    The fastest general purpose integer factorization algorithm known today is the General Number Field Sieve, and it still uses this idea from Dixon's method fundamentally. The improvements come from more efficient discovery of smooth numbers, by extending the factor bases over rings other than $mathbbZ$.






                    share|cite|improve this answer














                    The idea of representing integers as vectors of their prime factorization exponents is used in some of the fastest factorization algorithms which are based on Dixon's method. This trick allows turning the problem of finding a nontrivial factorization of an integer $n$ into a problem of linear algebra over $mathbbF_2$.



                    You first choose a set of prime numbers called the factor base (for example it can be the set of all prime numbers less than a bound $B$). Then you look for lots of numbers $x$ such that $x^2 pmod n$ is $B$-smooth (number that is divisible only by primes less than $B$). You can represent these numbers as vectors in terms of the exponents of the primes occuring in the prime factorization. We only consider the parity (exponent mod $2$) for reasons I'll explain later. The vectors live in a finite-dimensional vector space over $mathbbF_2$ with dimension equal to the size of the factor base.



                    Once you have found sufficiently many $B$-smooth numbers ($1$ more than the dimension of the vector space), by general linear algebra you are guaranteed that you have a nontrivial linear dependence among the vectors, and this in turn allows you to produce a nontrivial congruence $x^2 equiv y^2 pmod n$ by combining the relations. The reason we work over $mathbbF_2$ is because we only care about perfect squares, which means that all exponents in the prime factorization should be even.



                    With such a congruence of squares, you can use the identity $x^2 - y^2 = (x+y)(x-y)$ and with a high probability, this leads to a nontrivial factorization of $n$ by considering $gcd(n, x + y)$ or $gcd(n, x - y)$ (not guaranteed, but if you do that a couple times it will likely work).



                    The fastest general purpose integer factorization algorithm known today is the General Number Field Sieve, and it still uses this idea from Dixon's method fundamentally. The improvements come from more efficient discovery of smooth numbers, by extending the factor bases over rings other than $mathbbZ$.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited Dec 20 '18 at 22:31

























                    answered Dec 20 '18 at 22:16









                    Tob Ernack

                    2,229418




                    2,229418





















                        0














                        Along with that great data science rule of "graph early", I suggest starting with 2, 3, 5, and 7, i.e. the first four dimensions, and making animated 3D plots showing the allowable coordinates for some example numbers and to see what you get. So:



                         2 3 5 7
                        first plot (trivial)
                        1 0 0 0 0

                        second plot
                        2 1 0 0 0
                        2 0 x 0 0
                        2 0 0 y 0
                        2 0 z1 z2 0

                        third plot
                        2.5 . . . .
                        2.5 . . . .
                        2.5 . . . .





                        share|cite|improve this answer

























                          0














                          Along with that great data science rule of "graph early", I suggest starting with 2, 3, 5, and 7, i.e. the first four dimensions, and making animated 3D plots showing the allowable coordinates for some example numbers and to see what you get. So:



                           2 3 5 7
                          first plot (trivial)
                          1 0 0 0 0

                          second plot
                          2 1 0 0 0
                          2 0 x 0 0
                          2 0 0 y 0
                          2 0 z1 z2 0

                          third plot
                          2.5 . . . .
                          2.5 . . . .
                          2.5 . . . .





                          share|cite|improve this answer























                            0












                            0








                            0






                            Along with that great data science rule of "graph early", I suggest starting with 2, 3, 5, and 7, i.e. the first four dimensions, and making animated 3D plots showing the allowable coordinates for some example numbers and to see what you get. So:



                             2 3 5 7
                            first plot (trivial)
                            1 0 0 0 0

                            second plot
                            2 1 0 0 0
                            2 0 x 0 0
                            2 0 0 y 0
                            2 0 z1 z2 0

                            third plot
                            2.5 . . . .
                            2.5 . . . .
                            2.5 . . . .





                            share|cite|improve this answer












                            Along with that great data science rule of "graph early", I suggest starting with 2, 3, 5, and 7, i.e. the first four dimensions, and making animated 3D plots showing the allowable coordinates for some example numbers and to see what you get. So:



                             2 3 5 7
                            first plot (trivial)
                            1 0 0 0 0

                            second plot
                            2 1 0 0 0
                            2 0 x 0 0
                            2 0 0 y 0
                            2 0 z1 z2 0

                            third plot
                            2.5 . . . .
                            2.5 . . . .
                            2.5 . . . .






                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Dec 20 '18 at 19:57









                            elliot svensson

                            1011




                            1011












                                Popular posts from this blog

                                How to check contact read email or not when send email to Individual?

                                Displaying single band from multi-band raster using QGIS

                                How many registers does an x86_64 CPU actually have?