All Questions
Tagged with sums-of-squares prime-numbers
12 questions
15
votes
4
answers
2k
views
Square roots and prime numbers
Definitions:
Here I present a novel conjecture using basic mathematical tools like the sum of the
divisors of an integer $n$ called $\sigma(n)$, the sum of the squares of the positive divisors of n ...
- 127
11
votes
2
answers
614
views
Jacobi symbols for two-square sums of primes
Given a prime $p\equiv 1\pmod 4$, Fermat's two-squares theorem discovered by Girard
states that there exists two integers $A,B$ such that
$p=A^2+B^2$.
For all primes up to $10^7$ the integers $A$ and $...
- 17.9k
7
votes
2
answers
2k
views
Sums of squares of primes [closed]
Question: What is the least number that is a sum of three squares of primes in exactly six ways?
... I know it is not research mathematics. Happy new year!
EDIT: Now that it is answered I should ...
- 25.5k
5
votes
0
answers
229
views
Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?
As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares.
To avoid such congruence ...
- 15.6k
8
votes
1
answer
416
views
Every odd integer greater than $1$ is of the form $a+b$ with $a^2+b^2$ being prime
Let $m$ be an odd integer greater than $1$. Is it true that there are positive $a, b$ such that $m=a+b$ and $a^2+b^2$ is a prime number?
It seems that for every odd $m$ there are many $(a,b)\in \...
- 3,866
0
votes
1
answer
612
views
Number of fixed points in Zagier's involution (Fermat's Theorem) [closed]
Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ having ...
- 111
21
votes
1
answer
1k
views
Primes that are sums of two squares with constraints on the squares
It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are ...
- 213
5
votes
1
answer
727
views
Sum of two squares and implication of Bunyakovsky conjecture
Bunyakovsky's conjecture states that a polynomial with integer
coefficients takes infinitely many prime values at integers,
unless this is impossible for trivial reasons.
Let $a_1(x), a_2(x), a_3(x), ...
- 25.4k
7
votes
2
answers
426
views
Divisor sums over values of binary forms of primes
Let $\tau$ be the divisor function, that is
$$
\tau(n)=\sharp\{d \in \mathbb{N}, d|n\}.
$$
I was wondering if anyone has ever proved an asymptotic estimate
for the sum
$$S(x):=\sum_{p,q\leq x}\tau(p^...
- 3,062
6
votes
1
answer
653
views
On permuted sum of squares of primes in a list
We want to pick a set of distinct primes (if not possible, then just positive numbers) $p_1,p_2,\dots,p_k$ such that there exists $t$ permutations, $\sigma_1(\cdot)$,$\sigma_2(\cdot),\dots,\sigma_t(\...
- 13.9k
1
vote
0
answers
255
views
Confirm/refute $f(x)$ where $f(x) = $x-th Mersenne prime ($M_p$) where $x$ is [1,2,3,4,5,6,7...]
While tinkering with numbers, I found $n=f(x)$ where $n = \sigma(\sigma(n)-n)$ and $x \neq p$ where $p$ is a Mersenne prime exponent, but I need help input in regard to improving accuracy.
First off, ...
- 131
3
votes
3
answers
958
views
solutions to equation mod a prime
I know that characterizing the solutions to an equation in a finite field is generally difficult, but I was wondering if anyone had anything to say about the equation
(ab)^2 + a^2 + b^2 = 0 mod p
I ...
- 39