All Questions
Tagged with integer-partitions prime-numbers
4
questions
18
votes
6
answers
9k
views
Prime Partition
A prime partition of a number is a set of primes that sum to the number. For instance, {2 3 7} is a prime partition of $12$ because $2 + 3 + 7 = 12$. In fact, there ...
- 587
7
votes
1
answer
191
views
Finding $z=x+y$ such that $x^2 + y^2$ is prime
For which integers $z$ can one write $z=x+y$ such that $x^2+y^2$ is prime?
It feels like it should be possible for all odd $z>1$, and I have tried to adapt Euler's proof of Girard/Fermat's ...
- 71
6
votes
2
answers
241
views
$p\equiv 1\pmod 4\Rightarrow p=a^2+b^2$ and $p\equiv 1\pmod 8\Rightarrow p=a^2+2b^2$, what about for $p\equiv 1\pmod {2^n}$ in general
Primes $p$ with $p\equiv 1\pmod 4$ can be written as $p=a^2+b^2$ for some integers $a,b$. For $p\equiv 1\pmod 8$ we have $p=a^2+2b^2$. Can primes that satisfy $p\equiv 1\pmod{2^n}$ for $n>3$ be ...
- 1,950
2
votes
1
answer
92
views
MacMahon partition function and prime detection (ref arXiv:2405.06451)
In the recent paper arXiv:2405.06451 the authors provide infinitely many characterizations of the primes using MacMahon partition functions: for $a>0$ the functions $M_a(n):=\sum\limits_{0<s_1&...
- 747