sum of square is equal to prime
$begingroup$
Let $n in mathbb{N}$ and $p$ a prime number then I would like to find all $(a_1, ..., a_n) in mathbb{N}^n$ such that :
$$sum a_i^2 = p$$
Using a computer program It seems that the only solutions are $(1,...,1)$.
I don't see how to prove this result. An idea could be to look at the whole expression mod $4$, but it doesn't seem to help... :(
number-theory
asked Dec 11 '18 at 15:49
hdfjhqzuifqhhdfjhqzuifqh
1
$endgroup$
add a comment |
$begingroup$
Let $n in mathbb{N}$ and $p$ a prime number then I would like to find all $(a_1, ..., a_n) in mathbb{N}^n$ such that :
$$sum a_i^2 = p$$
Using a computer program It seems that the only solutions are $(1,...,1)$.
I don't see how to prove this result. An idea could be to look at the whole expression mod $4$, but it doesn't seem to help... :(
number-theory
asked Dec 11 '18 at 15:49
hdfjhqzuifqhhdfjhqzuifqh
1
$endgroup$
4
$begingroup$
There are lots of examples...$1^2+2^2=5$, say. Or $2^2+3^2=13$. Did you have some specific thing in mind?
$endgroup$
– lulu
Dec 11 '18 at 15:51
1
$begingroup$
Fermat's two square Theorem is relevant.
$endgroup$
– lulu
Dec 11 '18 at 15:53
$begingroup$
See OEIS sequence A001156 for the number of partitions of a given positive integer into squares.
$endgroup$
– Robert Israel
Dec 11 '18 at 16:12
add a comment |
$begingroup$
Let $n in mathbb{N}$ and $p$ a prime number then I would like to find all $(a_1, ..., a_n) in mathbb{N}^n$ such that :
$$sum a_i^2 = p$$
Using a computer program It seems that the only solutions are $(1,...,1)$.
I don't see how to prove this result. An idea could be to look at the whole expression mod $4$, but it doesn't seem to help... :(
number-theory
asked Dec 11 '18 at 15:49
hdfjhqzuifqhhdfjhqzuifqh
1
$endgroup$
Let $n in mathbb{N}$ and $p$ a prime number then I would like to find all $(a_1, ..., a_n) in mathbb{N}^n$ such that :
$$sum a_i^2 = p$$
Using a computer program It seems that the only solutions are $(1,...,1)$.
I don't see how to prove this result. An idea could be to look at the whole expression mod $4$, but it doesn't seem to help... :(
number-theory
number-theory
asked Dec 11 '18 at 15:49
hdfjhqzuifqhhdfjhqzuifqh
1
asked Dec 11 '18 at 15:49
hdfjhqzuifqhhdfjhqzuifqh
1
asked Dec 11 '18 at 15:49
hdfjhqzuifqhhdfjhqzuifqh
1
asked Dec 11 '18 at 15:49
hdfjhqzuifqhhdfjhqzuifqh
1
asked Dec 11 '18 at 15:49
hdfjhqzuifqhhdfjhqzuifqh
1
1
4
$begingroup$
There are lots of examples...$1^2+2^2=5$, say. Or $2^2+3^2=13$. Did you have some specific thing in mind?
$endgroup$
– lulu
Dec 11 '18 at 15:51
1
$begingroup$
Fermat's two square Theorem is relevant.
$endgroup$
– lulu
Dec 11 '18 at 15:53
$begingroup$
See OEIS sequence A001156 for the number of partitions of a given positive integer into squares.
$endgroup$
– Robert Israel
Dec 11 '18 at 16:12
add a comment |
4
$begingroup$
There are lots of examples...$1^2+2^2=5$, say. Or $2^2+3^2=13$. Did you have some specific thing in mind?
$endgroup$
– lulu
Dec 11 '18 at 15:51
1
$begingroup$
Fermat's two square Theorem is relevant.
$endgroup$
– lulu
Dec 11 '18 at 15:53
$begingroup$
See OEIS sequence A001156 for the number of partitions of a given positive integer into squares.
$endgroup$
– Robert Israel
Dec 11 '18 at 16:12
4
4
$begingroup$
There are lots of examples...$1^2+2^2=5$, say. Or $2^2+3^2=13$. Did you have some specific thing in mind?
$endgroup$
– lulu
Dec 11 '18 at 15:51
$begingroup$
There are lots of examples...$1^2+2^2=5$, say. Or $2^2+3^2=13$. Did you have some specific thing in mind?
$endgroup$
– lulu
Dec 11 '18 at 15:51
1
1
$begingroup$
Fermat's two square Theorem is relevant.
$endgroup$
– lulu
Dec 11 '18 at 15:53
$begingroup$
Fermat's two square Theorem is relevant.
$endgroup$
– lulu
Dec 11 '18 at 15:53
$begingroup$
See OEIS sequence A001156 for the number of partitions of a given positive integer into squares.
$endgroup$
– Robert Israel
Dec 11 '18 at 16:12
$begingroup$
See OEIS sequence A001156 for the number of partitions of a given positive integer into squares.
$endgroup$
– Robert Israel
Dec 11 '18 at 16:12
add a comment |
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4
$begingroup$
There are lots of examples...$1^2+2^2=5$, say. Or $2^2+3^2=13$. Did you have some specific thing in mind?
$endgroup$
– lulu
Dec 11 '18 at 15:51
1
$begingroup$
Fermat's two square Theorem is relevant.
$endgroup$
– lulu
Dec 11 '18 at 15:53
$begingroup$
See OEIS sequence A001156 for the number of partitions of a given positive integer into squares.
$endgroup$
– Robert Israel
Dec 11 '18 at 16:12