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# Questions tagged [sums-of-squares]

For questions concerning various representation of integers as sums of squares, which are studied in number theory.

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### Primality test using the sum of 2 squares representation of a prime $p^2= a^2+b^2$

We have seen that the sum of two squares (2 sq rep) of an integer $N^2$ can be used to factor $N$. Can the sum of two squares be used to factor large numbers? Here we use the same method to show ...
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### Find the next least number N that is N+1 = X^2 and (N/2)+1 = y^2?

To go with this following task to create a proper equation ? Many numbers, especially from smaller ones, can rightly claim that they are of particular interest to scientists. In the kingdom of ...
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### Can the sum of two squares be used to factor large numbers?

The method described below is different from Euler sum of two squares factorization method. The method below does not require 2 sum of two squares representations to factor a number. This post was ...
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### Can the sum of two squares be used to determine if a number is square free?

Mathword (http://mathworld.wolfram.com/Squarefree.html) stated that "There is no known polynomial time algorithm for recognizing squarefree integers or for computing the squarefree part of an integer."...
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### If $xy$ divides $x^2 + y^2$ show that $x=\pm y$ [duplicate]

Problem statement : Let $x,y$ be integers, show that if $xy$ divides $x^2 + y^2$ then $x=\pm y.$ What I have tried: I can reduce this to the case where $\gcd(x,y)=1$, since if $x$ and $y$ have a ...
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### When is $n^2-1$ a sum of two squares?

I am trying to work out when $n^2-1$ is a sum of two squares. Is there a formula for such $n$? I have found $n=1$, $n=3$ and $n=9$ so far but am struggling to find a pattern that will generalise. If ...
which primes $\mathfrak{p}$ can be written as sums of two squares in $\mathbb{Q}(\sqrt{2})$? I am asking to solve an equation: $$\mathfrak{p} = \big(a_1+a_2\sqrt{2}+a_3\sqrt{4}\big)^2 + \... 1answer 58 views ### A Peculiar Sum Of Squares [duplicate] I have been trying to solve a question that arose in my mind a couple of days ago. What is the sum of: 1+4+9+16+25+\cdots +n^2? It is the sum of squares of each numbers starting from 1 to n.... 1answer 49 views ### New Figurate Number Relation? [closed] Has anyone seen the relation$$ nP_{2}(n)=P_{3}(n-1)+\sum_{i=1}^ni^2 $$where P_2(n) is the nth triangular number and P_3(n) is the nth tetrahedral number? I know the straightforward algebra ... 0answers 38 views ### why sum of squares equals to assign the variable evenly in linear programming I have a quesiton about linear programming. My objective is trying to make the utilization evenly. Example U_{1}, U_{2},...,U_{k}  \sum_{i=1..k} U_{i} = C  C is some constant. U_{i} is ... 2answers 143 views ### Product of (4k-1) primes can't be sum of 2 squares I am trying to prove, Product of primes of the form (4k-1) can't be sum of 2 squares. My approach is- Let the product is M=m_1m_2...m_n where  m_1, m_2, ...m_n are primes. Assume, M can ... 2answers 131 views ### Integer as sum of 6 squares. Can every integer be written as sum of exactly 6 squares? I am also curious to know when an integer can be written as sum of exactly 8 squares. I know the problem is related to Waring-Hilbert ... 1answer 37 views ### How can be proved that every square number can be expressed as the sum of another square number and some semiprime number? [closed] Any help to focus the problem would be welcomed! Something to do with quadratic residues? 1answer 24 views ### What the difference is between TypeI/II/III SS in ANOVA? My background is not mathematics and I do not really understand what this mathematical symbols means: Type I SS: SS(A) SS(B|A) SS (AB|A, B) Type II SS: SS(A|B) SS(B|A) SS (AB|A, B) Type III SS: SS (A|... 2answers 65 views ### Does \exists n : 6 + \sum_{i=2}^n p_i = x^6+y^6 for p_i the i^{\text{th}} prime and x,y\in\mathbb{Z}? I noticed something about the prime numbers: Pick the number 2. Then, add the first odd prime, namely 3. The result is 5 = 1^2+2^2. Notice that the exponents are also the number we picked. ... 5answers 150 views ### How do I find the sum of the series -1^2-2^2+3^2+4^2-5^2… upto 4n terms? [duplicate] I tried by giving$$ S = \sum_{k=0}^{n-1} \left((4k+3)^2+(4k+4)^2-(4k+1)^2-(4k+2)^2\right) $$but I am stuck here. I have no idea what to do next. The answer in my book says 4n(n+1). How can I get ... 1answer 120 views ### Does every sum-of-squares equation have a plane geometric interpretation? The 2.1.2 sum-of-squares [SOS] equation$$a^2=b^2+c^2$$can be thought of in the [regular XY] plane as a right-angled triangle. Question: Is there an analogous interpretation for every SOS equation?... 1answer 48 views ### Confusion about sum of squares. First, I want to declare that this problem have been solved by myself, I just want to share my work to everyone. It has known that a prime has the form 4k+3 cannot be sum of two squares, and also a ... 2answers 369 views ### Sum of two squares equal to 2018^{2019}+2018 [closed]$$x^2+y^2 = 2018^{2019}+2018$$is expressed as sum of two perfect squares. Any pair of perfect squares can satisfy? 1answer 85 views ### Find all triples (x,y,z) of positive integers such that 2018^x=y^2+z^2+1 [closed] Find all triples (x,y,z) of positive integers such that$$2018^x=y^2+z^2+1$$1answer 275 views ### For any integer n, 78n^2+72n+2018 is expressed as sum of four perfect cubes. [closed] For any integer n,$$78n^2+72n+2018$$is expressed as sum of four perfect cubes. Is this possible that any integer can fit into this expression with 4 perfect cubes? 1answer 214 views ### Is it possible to use partitions of an odd integer to generate primes in a given interval? We start with the partition of N=5.$$54+13+23+1+12+2+12+1+1+11+1+1+1+1$$Then we form the sum of squares (no limit on the number of elements) to get:$$4^2+1^2=17$$... 1answer 118 views ### Find sum of squares of elements of QB Given that Q is an orthogonal nxn matrix and B is an mxn matrix, how can we find the sum of squares of all elements of QB in terms of the sum of squares of all elements of B? I know that the sum of ... 1answer 168 views ### Mandelbrot and Julia fractals for z_{n+2} = z_{n+1}^2 + z_n^2 + c The Mandelbrot and Julia type fractals are very Well known. But such fractals follow from$$z_n = f(z_{n-1},c) In other words a recursion that only depends on the previous value and a constant. (...
I refer to this question. Given a perfect square, can you prove that it is a sum of two perfect squares? I recently saw this: Let $p,q$ be primes. $p_i \equiv 1 \pmod 4$ and $q_i \equiv 3 \pmod 4$...