Differences between numbers of the form abcdef and acdef which are perfect square.
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$541456-51456=700^2$ is the example I found. Are there other examples of numbers of the form abcdef such that abcdef-acdef is a perfect square?
number-theory
asked 14 hours ago
homunculus
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up vote
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$541456-51456=700^2$ is the example I found. Are there other examples of numbers of the form abcdef such that abcdef-acdef is a perfect square?
number-theory
asked 14 hours ago
homunculus
393
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homunculus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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up vote
2
down vote
favorite
up vote
2
down vote
favorite
$541456-51456=700^2$ is the example I found. Are there other examples of numbers of the form abcdef such that abcdef-acdef is a perfect square?
number-theory
asked 14 hours ago
homunculus
393
New contributor
homunculus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$541456-51456=700^2$ is the example I found. Are there other examples of numbers of the form abcdef such that abcdef-acdef is a perfect square?
number-theory
number-theory
asked 14 hours ago
homunculus
393
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homunculus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 14 hours ago
homunculus
393
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homunculus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 14 hours ago
homunculus
393
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homunculus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 14 hours ago
homunculus
393
asked 14 hours ago
homunculus
393
393
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homunculus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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2 Answers
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up vote
4
down vote
$$abcdef-acdef=10^4b+10^5a-(10^4a)=10^4(9a+b)$$
So, all we need is $9a+b$ to be perfect square
answered 14 hours ago
lab bhattacharjee
219k14153268
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up vote
1
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Of course,
$$
beginaligned
overlineabcdef - overlineacdef
&=
overlineab0000 - overlinea0000
\
&=10000cdot(overlineab - overlinea)
\
&=100^2cdot(9a+b) .
endaligned
$$
So we can choose $c,d,e,f$ arbitrarily, and for $a,b$ there are the following chances:
for a in [1..9]:
for b in [0..9]:
ab_a = 9*a + b
if ab_a.is_square():
print ( "a=%s b=%s %s-%s = %s^2"
% (a, b, 10*a+b, a, sqrt(ab_a)) )
And the results are:
a=1 b=0 10-1 = 3^2
a=1 b=7 17-1 = 4^2
a=2 b=7 27-2 = 5^2
a=3 b=9 39-3 = 6^2
a=4 b=0 40-4 = 6^2
a=5 b=4 54-5 = 7^2
a=7 b=1 71-7 = 8^2
a=8 b=9 89-8 = 9^2
a=9 b=0 90-9 = 9^2
Here i was using sage.
answered 14 hours ago
dan_fulea
5,7251312
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
$$abcdef-acdef=10^4b+10^5a-(10^4a)=10^4(9a+b)$$
So, all we need is $9a+b$ to be perfect square
answered 14 hours ago
lab bhattacharjee
219k14153268
add a comment |
up vote
4
down vote
$$abcdef-acdef=10^4b+10^5a-(10^4a)=10^4(9a+b)$$
So, all we need is $9a+b$ to be perfect square
answered 14 hours ago
lab bhattacharjee
219k14153268
add a comment |
up vote
4
down vote
up vote
4
down vote
$$abcdef-acdef=10^4b+10^5a-(10^4a)=10^4(9a+b)$$
So, all we need is $9a+b$ to be perfect square
answered 14 hours ago
lab bhattacharjee
219k14153268
$$abcdef-acdef=10^4b+10^5a-(10^4a)=10^4(9a+b)$$
So, all we need is $9a+b$ to be perfect square
answered 14 hours ago
lab bhattacharjee
219k14153268
answered 14 hours ago
lab bhattacharjee
219k14153268
answered 14 hours ago
lab bhattacharjee
219k14153268
answered 14 hours ago
lab bhattacharjee
219k14153268
219k14153268
add a comment |
add a comment |
up vote
1
down vote
Of course,
$$
beginaligned
overlineabcdef - overlineacdef
&=
overlineab0000 - overlinea0000
\
&=10000cdot(overlineab - overlinea)
\
&=100^2cdot(9a+b) .
endaligned
$$
So we can choose $c,d,e,f$ arbitrarily, and for $a,b$ there are the following chances:
for a in [1..9]:
for b in [0..9]:
ab_a = 9*a + b
if ab_a.is_square():
print ( "a=%s b=%s %s-%s = %s^2"
% (a, b, 10*a+b, a, sqrt(ab_a)) )
And the results are:
a=1 b=0 10-1 = 3^2
a=1 b=7 17-1 = 4^2
a=2 b=7 27-2 = 5^2
a=3 b=9 39-3 = 6^2
a=4 b=0 40-4 = 6^2
a=5 b=4 54-5 = 7^2
a=7 b=1 71-7 = 8^2
a=8 b=9 89-8 = 9^2
a=9 b=0 90-9 = 9^2
Here i was using sage.
answered 14 hours ago
dan_fulea
5,7251312
add a comment |
up vote
1
down vote
Of course,
$$
beginaligned
overlineabcdef - overlineacdef
&=
overlineab0000 - overlinea0000
\
&=10000cdot(overlineab - overlinea)
\
&=100^2cdot(9a+b) .
endaligned
$$
So we can choose $c,d,e,f$ arbitrarily, and for $a,b$ there are the following chances:
for a in [1..9]:
for b in [0..9]:
ab_a = 9*a + b
if ab_a.is_square():
print ( "a=%s b=%s %s-%s = %s^2"
% (a, b, 10*a+b, a, sqrt(ab_a)) )
And the results are:
a=1 b=0 10-1 = 3^2
a=1 b=7 17-1 = 4^2
a=2 b=7 27-2 = 5^2
a=3 b=9 39-3 = 6^2
a=4 b=0 40-4 = 6^2
a=5 b=4 54-5 = 7^2
a=7 b=1 71-7 = 8^2
a=8 b=9 89-8 = 9^2
a=9 b=0 90-9 = 9^2
Here i was using sage.
answered 14 hours ago
dan_fulea
5,7251312
add a comment |
up vote
1
down vote
up vote
1
down vote
Of course,
$$
beginaligned
overlineabcdef - overlineacdef
&=
overlineab0000 - overlinea0000
\
&=10000cdot(overlineab - overlinea)
\
&=100^2cdot(9a+b) .
endaligned
$$
So we can choose $c,d,e,f$ arbitrarily, and for $a,b$ there are the following chances:
for a in [1..9]:
for b in [0..9]:
ab_a = 9*a + b
if ab_a.is_square():
print ( "a=%s b=%s %s-%s = %s^2"
% (a, b, 10*a+b, a, sqrt(ab_a)) )
And the results are:
a=1 b=0 10-1 = 3^2
a=1 b=7 17-1 = 4^2
a=2 b=7 27-2 = 5^2
a=3 b=9 39-3 = 6^2
a=4 b=0 40-4 = 6^2
a=5 b=4 54-5 = 7^2
a=7 b=1 71-7 = 8^2
a=8 b=9 89-8 = 9^2
a=9 b=0 90-9 = 9^2
Here i was using sage.
answered 14 hours ago
dan_fulea
5,7251312
Of course,
$$
beginaligned
overlineabcdef - overlineacdef
&=
overlineab0000 - overlinea0000
\
&=10000cdot(overlineab - overlinea)
\
&=100^2cdot(9a+b) .
endaligned
$$
So we can choose $c,d,e,f$ arbitrarily, and for $a,b$ there are the following chances:
for a in [1..9]:
for b in [0..9]:
ab_a = 9*a + b
if ab_a.is_square():
print ( "a=%s b=%s %s-%s = %s^2"
% (a, b, 10*a+b, a, sqrt(ab_a)) )
And the results are:
a=1 b=0 10-1 = 3^2
a=1 b=7 17-1 = 4^2
a=2 b=7 27-2 = 5^2
a=3 b=9 39-3 = 6^2
a=4 b=0 40-4 = 6^2
a=5 b=4 54-5 = 7^2
a=7 b=1 71-7 = 8^2
a=8 b=9 89-8 = 9^2
a=9 b=0 90-9 = 9^2
Here i was using sage.
answered 14 hours ago
dan_fulea
5,7251312
answered 14 hours ago
dan_fulea
5,7251312
answered 14 hours ago
dan_fulea
5,7251312
answered 14 hours ago
dan_fulea
5,7251312
5,7251312
add a comment |
add a comment |
homunculus is a new contributor. Be nice, and check out our Code of Conduct.
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