Questions tagged [sums-of-squares]

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

108 questions with no upvoted or accepted answers
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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 ...
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29 views

Representations of integer in the form f(x) - f(y)?

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}. ...
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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 $$A_{ij}A_{ij} = 1$$ ...
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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 ...
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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 ...
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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*...
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160 views

Is integer factoring a combination of Fermat's sum of 4 and difference of 2 squares?

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 ...