Intereting Posts

Euler characteristic of an $n$-sphere is $1 + (-1)^n$.
Which holomorphic function is this the real part of?
Are isomorphic the following two links?
Is $|f(a) – f(b)| \leqslant |g(a) – g(b)| + |h(a) – h(b)|$? when $f = \max\{{g, h}\}$
How many positive integers $ n$ with $1 \le n \le 2500$ are prime relative to $3$ and $5$?
If two norms are equivalent on a dense subspace of a normed space, are they equivalent?
bijection between prime ideals of $R_p$ and prime ideals of $R$ contained in $P$
Does Chaitin's constant have infinitely many prime prefixes?
The constant of integration during integration by parts
Proving that $\mathbb{A}=\{\alpha \in \mathbb{C}: \alpha \text{ is algebraic over } \mathbb{Q} \}$ is not a finite extension
What's the difference between arccos(x) and sec(x)
Smash product of compact spaces
Find all polynomials that fix $\mathbb Q$ and the irrationals
Why is there no natural metric on manifolds?
what is genus of complete intersection for: $F_1 = x_0 x_3 – x_1 x_2 , F_2 = x_0^2 + x_1^2 + x_2^2 + x_3^2$

If n is a positive integer that can be represented as the sum of two odd squares in two different ways:

$$

n = a^2 + b^2 = c^2 + d^2

$$

where $a$, $b$, $c$ and $d$ are discrete odd positive integers, what properties can be deduced about $a$, $b$, $c$ and $d$? There’s lots of information online about what properties of $n$ are, and its prime factors, but I can’t find, or deduce myself, anything about $a$, $b$, $c$ and $d$. Are there any relationships between them?

- Finding pairs of triangular numbers whose sum and difference is triangular
- Square of four digit number $a$
- Exponential Diophantine equation $7^y + 2 = 3^x$
- Proving that the Calkin-Wilf tree enumerates the rationals.
- Find integers $(w, x, y, z)$ such that the product of each two of them minus 1 is square.
- Show that the sequence $1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,1,3,1,4,1,5,1,6,1,7,1,8,1,9,2,0,2,1,\cdots$ isn't periodic
- Congruent Modulo $n$: definition
- Is the set of all numbers which divide a specific function of their prime factors, infinite?
- 31,331,3331, 33331,333331,3333331,33333331 are prime
- Multiplicative Order of $b \pmod p$ , where $p \equiv 1 \pmod 4$

What you want is

A Note on Euler’s Factoring Problem

By: John Brillhart

This note consists of a brief introduction to Euler’s factoring problem and his results, as well as a complete and elegant solution to the problem given by Lucas and Matthews about a century later.

from the December 2009 Monthly, pages 928-931, see

http://www.maa.org/pubs/monthly_dec09_toc.html

Alright, I pasted in the pdf but the result was not entirely legible. Also there is some question of legality as the article is pretty recent. I can email the pdf to individuals who send me a request.

However, I can typeset Theorem 2:

Let $N>1$ be an odd integer expressed in two different ways as

$$ N = m a^2 + n b^2 = m c^2 + n d^2, $$

where $a,b,c,d,m,n \in \mathbb Z^+, \; b < d,$ and

$\gcd(ma,nb) = \gcd(mc,nd) =1.$ Then

$$ N = \gcd(N, ad-bc) \cdot \; \frac{N}{\gcd(N, ad-bc)} $$

where the factors are nontrivial.

Your case would be $m=n=1.$ Note that then if the theorem cannot be used because $\gcd(a,b) = g >1,$ then we have some factoring anyway, as $g^2 | N.$

Going to the Gaussian integers (the complex numbers of the form $x+yi$, where $x,y$ are ordinary integers, and $i=\sqrt{-1}$), we get $$(a+bi)(a-bi)=(c+di)(c-di)$$ Then there exist Gaussian integers $r,s$ such that $$a+bi=rs,\quad a-bi=\overline r\overline s,\quad c+di=r\overline s,\quad c-di=\overline rs$$ Note that $\overline r$ is the complex conjugate of $r$; if $r=x+yi$, then $\overline r=x-yi$. From this we get $$a=(rs+\overline r\overline s)/2,\quad b=(rs-\overline r\overline s)/2i,\quad c=(r\overline s+\overline rs)/2,\quad d=(r\overline s-\overline rs)/2i$$ You could possibly take this a step further by writing $r=x+yi$, $s=w+zi$ and multiplying everything out and combining like terms; I think that’s as close as you’ll get to relating $a,b,c,d$.

- Conditions for a unique root of a fifth degree polynomial
- Show that $\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx =-\frac{\pi^2 \sqrt{2}}{16}$
- Real-analytic $f(z)=f(\sqrt z) + f(-\sqrt z)$?
- definit integral of Airy function
- Find the sum of $-1^2-2^2+3^2+4^2-5^2-6^2+\cdots$
- Is $f(x,y)=(x+2y+y^2+|xy|, 2x+y+x^2+|xy|)$ differentiable?
- Other ways to deduce Cyclicity of the Units of certain groups?
- Finitely generated projective modules are isomorphic to their double dual.
- Degeneracy in Linear Programming
- Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$
- Equivalent characterizations of faithfully exact functors of abelian categories
- Subset of the preimage of a semicontinuous real function is Borel
- Prove the converges of the followin sequence and find the limit
- Evaluate the integral $\int_{0}^{+\infty}\left(\frac{x^2}{e^x-1}\right)^2dx$
- Simple group of order $660$ is isomorphic to a subgroup of $A_{12}$