# Questions tagged [sums-of-squares]

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

466 questions
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### Diophantine equation $a^2+b^2+c^2=a^2b^2$

I am trying to find all non trivial integers for which $a^2+b^2+c^2=a^2b^2$. As suggested I have tried working (mod 4). This is what I've gotten so far: Squares can have a remainder of 0 or 1 (mod 4)....
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### How many numbers up to $x$ are sums of two squares?

I know that the primes which are representable as a sum of two squares are a specific type of prime, that is, $p=4k+1$, where $k$ is a positive integer. and from this, I could deduce which integers ...
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### 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 $n$....
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### 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 $n$th triangular number and $P_3(n)$ is the $n$th tetrahedral number? I know the straightforward algebra ...
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### Sum of squares for truncated normal distribution

I need to find a distribution of sum of squares of N variables, when N-1 variable has a normal distribution and one variable has a truncated normal distribution. Should be similar to chi-square ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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|...
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### 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. ...
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### 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 ...
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### 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?...
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### 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 ...
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### 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?
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### 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 $$2018^x=y^2+z^2+1$$
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### 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?
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### Is it possible to use partitions of an odd integer to generate primes in a given interval?

We start with the partition of $N=5$. $$5$$ $$4+1$$ $$3+2$$ $$3+1+1$$ $$2+2+1$$ $$2+1+1+1$$ $$1+1+1+1+1$$ Then we form the sum of squares (no limit on the number of elements) to get: $$4^2+1^2=17$$ ...
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### 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 ...
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