# Questions tagged [sums-of-squares]

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

484 questions
Filter by
Sorted by
Tagged with
136 views

### Minimum over Probability Measures

Let $f$ and $g$ be polynomials in $\mathbf x \in \mathbb R^n$. Let $X$ be a compact subset of $\mathbb R^n$. Finally, Let $\mathcal M(X)$ be the set of probability measures over $X$. Can the ...
38 views

### Quadratic fit using sum of least squares without matrices

I've been hunting around for examples using the sum of least squares + partial derivative method to fit a polynomial to a set of points but am completely stuck. All the examples I've found involve ...
83 views

### $\forall n \in \mathbb{N}_{>5}\implies\exists (a,b)\in \mathbb Z^+:a^2+b^2\notin\mathbb P\;\wedge\;n=a+b$

Conjecture: $\forall n \in \mathbb{N}_{>5} \implies\exists (a,b)\in \mathbb Z^+:a^2+b^2\notin\mathbb P\;\wedge\;n=a+b$ Tested $\forall n\leq 100,000$. Small exceptions: {1,2,3,5}. I would like to ...
78 views

### Solving a Quadratic Ternary Form with Large Coefficients

I encountered this expression $$1215696x^2+566544y^2-103776z^2=0$$ which I've understood is called a "quadratic ternary form" and have been trying to find solutions $x,y,z\in\mathbb{Z}$ One can ...
1k views

### Value of $(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$ if $z^4-2z^3+z^2+z-7=0$ for $z=\alpha$, $\beta$, $\gamma$, $\delta$

Let $\alpha$, $\beta$, $\gamma$, $\delta$ be the roots of $$z^4-2z^3+z^2+z-7=0$$ then find value of $$(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$$ Are Vieta's formulas appropriate?
128 views

### How to prove that $441 \mid a^2 + b^2$

How to prove that $441 \mid a^2 + b^2$ if it is known that $21 \mid a^2 + b^2$. I've tried to present $441$ as $21 \cdot 21$, but it is not sufficient.
56 views

### Arithmetic progression of four square number with the same common difference [duplicate]

I want to know, if is possible found the arithmetic progression of four square number, with the same common difference. \begin{align} \ & a^2 +r = b^2 \\ & b^2 +r = c^2 \\ & c^2 +r =...
273 views

### Solve $32x^2 -y^2 = 448$

I am trying to find all integer solutions to the following equation: $$32x^2 - y^2 = 448$$ This is what I have tried so far: The equation describes a hyperbola, and so I try the usual trick of ...
1k views

### Can we prove a stronger claim?

Here it is asked whether for every $k>1$, there is a prime $p$ and $w_1,w_2,\cdots ,w_k>1$, such that $p$ divides $w_1^2+w_2^2+\cdots +w_k^2$, but none of the summands. Can we prove there is ...
143 views

### Primes dividing sums of squares but not dividing any of summands

I am inexperienced when it comes to number theory (even elementary one) so do not have an idea at this moment on how to solve this one. Let us go through some examples. For $k=2$ we can have prime ...
67 views

65 views

110 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 ...
100 views

69 views

### Any integer is of the form $n=a_1^2+a_2^2+a_3^2-a_4^2-a_5^2$

I've struggled with this conjecture, which probably can be proved: Any natural number $n$ can be written as $n=a_1^2+a_2^2+a_3^2-a_4^2-a_5^2$ for some $a_1,a_2,a_3,a_4,a_5\in\mathbb Z^+$. I guess ...
85 views

### Hard for-all-integer-problems proveable for very great integer limits

Computational experiments suggests the conjecture: All big enough odd numbers $N$ is of the form $N=\sum_{k=1}^n m_k$, where $\sum_{k=1}^n m_k^2$ is prime and all $m_k\in\mathbb Z^+$. Since the ...
351 views

### Yet an other conjecture about odd numbers $n=a+b$ such that $a^2+b^2$ is prime

This question is related to A conjecture about an unlimited path and Any odd number is of form $a+b$ where $a^2+b^2$ is prime but I present it on its own if anyone would like to help finding ...
194 views

### ANOVA: What are the values SSW, SSB and the test statistics?

In a test of the ability of a certain polymer to remove toxic wastes from water, experiments where conducted at three different temperatures. The data below give the percentages of impurities that ...
64 views

### Why $a^4 + b^4$ ($a, b$ are positive integers) is not a perfect square?

I've realized that there are maybe no positive integers $(a, b)$ that $a^4 + b^4$ is a perfect square, because I tested for $a, b \le 10000$ and cannot find any solution. I think that's weird, and ...