# 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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### Pattern in digits of sums of consecutive squares

I am interested in patterns in square numbers as well as the reasons behind them and I can't seem to figure out (also how to prove) why do the sums of two consecutive squares only end in digits 1, 3 ...
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### Algorithm Identification

Background I'm currently working with a system that has a 4-dimensional function. Currently, an algorithm is used to speed up calculation of the final value via interpolation, and two of the ...
Let $f(x) \in \mathbb{Z}[x]$ be a polynomial. I would like to have an estimate for the number of representations $R(n)$ of $n \in \mathbb{Z}$ in the form $$f(x) - f(y) = n, \qquad x,y \in \mathbb{N}. ... 0answers 113 views ### Square matrix whose sum of squared elements equals 1. I'm doing some applied work where I've come across examples that involve real valued square matrices that hold the following property, which expressed using tensor notation is$$A_{ij}A_{ij} = 1$$... 0answers 104 views ### Reduce (a+b)^2(c+d)^2-16abcd to sum of squares Is it possible to reduce$$(a+b)^2(c+d)^2-16abcd$$to sum of squares? This expression is used in proof of AM-GM inequality. It is known that$$\dfrac{a+b}{2}\ge\sqrt{ab}\tag{1}$$So, it can be proved ... 2answers 110 views ### Represent a prime number p congruent to 1 \pmod{3} by a sum of a square and 3 times a square I want to have a proof of the fact that each prime number is the sum of a square and three times a square (Euler). Context I read the answer to my former question about the number of points on an ... 2answers 114 views ### sum of square derivative What is the partial derivative of the following expression with respect to U_i,V_j and M, respectively:$$L=\sum_{i}^m \sum_{j}^n(P_{ij} - g(U_i^T M V_j))^2 $$where$$ U \in R^{d*m} , V \in R^{d*...
By sum of squares is meant the representation of an integer as a sum of 4 squares ( or five in some special cases ). By difference of squares is meant the usual $$x^2-y^2=(x-y)\cdot (x+y)$$ Here we ...