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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7
votes
1answer
532 views

Can the sum of the first n squares of primes be a perfect square?

So far, all I have found is for $n=1$, for which the square is 4. (Rather trivial) To give some context, I recently watched the Numberphile video Squared Squares. Basically, a squared square is a ...
2
votes
0answers
104 views

Sums of more than one combination of squares.

I'm interested in examples like these, where the sum of $n$ squares equals the sum of another $n$ squares. $3^2 + 11^2 \quad = \quad 7^2 + 9^2 \quad = \quad 130$ $5^2 + 6^2 + 10 ^ 2 \quad = \quad 4^...
1
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0answers
64 views

Given a closed form for a series, what can be said about the sum of the squares of its terms?

Suppose I have an integer sequence $\{a_k\}$, and suppose I know a closed form in terms of $n$ for this sum: $$\displaystyle\sum\limits_{k=1}^{n} a_k$$ Given this, is it always (or ever) possible to ...
0
votes
0answers
44 views

Can someone prove this? Difference between two exponents sum

$$y^n-x^m-\int^{n}_{m}x^\phi\ln{x}\text{ d}\phi\doteq\sum^{n}_{i=1}\sum^i_{\rho=0}\left((-1)^\rho\cdot\frac{\prod\limits_{j_1=\rho+1}^{i}(j_1)}{(i-\rho)!y^\rho}y^ix^\rho\right)\frac{\prod\limits_{j_2=...
11
votes
1answer
152 views

Sums of squares (Proof) [duplicate]

Prove that $n^2 + (n + 1)^2 = m^3$ does not have solutions in the positive integers. I guess that the proof is by contradiction, but if I suppose it, I can't find the contradiction. Thanks for your ...
0
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1answer
44 views

How to find $\cot (x)$ of this square?

So according to question, what values of $\cot (x)$? So I've tried and got $\frac{-1}{4}$ Thanks for helping!
1
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1answer
219 views

Express $k$ as the sum of three square numbers

How do you express $k$ as the sum of three square numbers, if $$m^2 + 3 = 2k$$ where both $m$ and $k$ are integers (both positive or negative if possible). It is known that $m$ must be an odd ...
1
vote
1answer
153 views

Finding other representations for a sum of two squares

For a sum of two squares: $a^2 + b^2 = c$ where $0 < b < a$, if I know of a valid $a^2$ and $b^2$, is there a fast algorithm for finding all other two square combinations that have the same sum? ...
4
votes
1answer
365 views

$x^2+y^2=-1 \mod p$ is solvable if $p \equiv 3 \mod 4$

Is there an elementary proof (i.e. without using fields and norms) for the fact that for a prime $p$ the congruence $x^2+y^2=-1 \mod p$ is solvable if $p \equiv 3 \mod 4$?
1
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2answers
80 views

The product of a sum of squares is a sum of squares [duplicate]

If $x=a^2+b^2$ and $y=c^2+d^2$ how can i prove that xy is also the sum of two rational squares? My teacher told me there are various methods to attack this problem but an easy way is to use the ...
1
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0answers
96 views

Product of SOS polynomials in the Gram matrix form by odd degree monomials.

This might be elementary, but I'm struggling to write the product of a sums of squares polynomial and an odd power of a variable using its Gram matrix. Following Pablo Parrilo's notation, let $p(x,y):...
3
votes
3answers
300 views

Math Problem with the sum of squares in a big loop

I have the following math problem (writen in code): ...
0
votes
1answer
30 views

Show that there are infinitely many solutions $(t, d)$ to $c - d^2 = t^2$ where $2c-1 = 3t^2$.

Set $c = (3t^2+1)/2$. Prove that infinitely many times, $c - t^2 = d^2$ for positive integers $t, d$. The first such instance is with $t = 1, d = 1$, and $t = 7, d = 5$. Can anyone come up with more ...
9
votes
2answers
463 views

How do I prove that powers of 5 are the sum of two squares using mathematical induction [duplicate]

Using mathematical induction I need to prove $$5^n=a_n^2 +b_n^2$$ What I've tried P(1): $$5^1 = 1^2 +2^2$$ Which is true P(2): $$5^2 = 3^2 +4^2$$ Which is also true Now for the induction ...
3
votes
1answer
94 views

Find the number of positive integer solutions to the equation $a^2+b^2 = p_1p_2p_3$

Find the number of positive integer solutions to the equation $$a^2+b^2 = p_1p_2p_3 $$ where the $p_i$ are distinct primes each congruent to $1$ mod $4$. My take: We can show each $p_i$ can be ...
0
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0answers
70 views

Problem on Natural numbers and perfect squares

What is the smallest natural number to be added to the natural number $n$ to make it a perfect square?
7
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2answers
956 views

Pythagorean Quadruples:

Consider the set of integers $x_1, x_2, x_3, x_4$ Such that: $$x_1^2 + x_2^2 + x_3^2 = x_4^2$$ How does one compute all the solutions to this system? I have the following method in place for ...
2
votes
8answers
296 views

Finite sum $\sum_{j=0}^{n-1} j^2$

How can I calculate this finite sum? Can someone help me? $$\sum_{j=0}^{n-1} j^2$$
2
votes
1answer
151 views

Sum of squares by treating it as a nonhomogenous recurrence

Let $T(n) = T(n-1) + n^2$ where $T(0) = 0$. The homogenous part $T(n) = T(n-1)$ has characteristic polynomial $x - 1 = 0$ and root $1$, which means $T(n) = \alpha \cdot 1^n$ for the homogenous part. ...
1
vote
1answer
59 views

For $a,b \in \sum_2$, prove $ab \in \sum_2$

I believe this is a basic proof but just hoping someone could give me some feedback on my attempt. $\sum_2:=\{x^2+y^2 | x,y \in \mathbb{Z}\}$ Since $a,b \in \sum_2$ it follows: $a=u^2+v^2$ $b=n^2+...
2
votes
2answers
527 views

Sum of two squares and prime factorizations

If $m$ is a positive integer such that every prime factor of $m$ that is congruent to $3$ modulo $4$ appears with an even power, then $m$ can be written as a sum of two squares. I wrote $m=2^{a_0}p_1^...
0
votes
1answer
53 views

Is the difference between consecutive sums of N consecutive squares trivial or already a theorem?

I noticed that the difference between two consecutive sums of N (integer) consecutive squares is equal to 2 x N. Is this trivial or already known and generalisable for all N? Not sure if this is ...
1
vote
1answer
114 views

“the bigger RegSS, the better the estimated model performs” are we sure?

Wikipedia here says "the bigger ESS (or RegSS, SSR, etc), the better the estimated model performs". Surely this is wrong. ESS is the sum of the squared distances between the fitted values and the ...
1
vote
3answers
388 views

Deciding if numbers can be written as the sum of three squares

I am asked to decide if 154, 155 and 156 can be written as the sum of three squares. I am using the theorem that if $n\in S_3$ then $n\not= 4^e(8k+7)$. Now just looking at all of these numbers it ...
0
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0answers
389 views

A simple algorithm for representing a square of the form $4k+1$ as a sum of 3 squares

Let's start with an example, $N=n^2=7^2=49$. $49$ can also be produced by simply writing $49= (6+1)^2 = 36 + 2\cdot6 + 1 = 36 + 13$. So if we can find a way to represent 13 as a sum of two squares, we ...
1
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0answers
35 views

Can someone show me a simple example of finding simple squared error in k-means?

I'm working on some homework and one of our assignments asks us to find the final mean of each cluster and the SSE (sum of squared error). I haven't been able to find anything online and was wondering ...
3
votes
0answers
80 views

Express Integer as the sum of $k$ squares

I'm looking for a way to efficiently determine the solutions ( i.e. sets of $x_1^2, x_2^2, ..., x_k^2 $ ) that satisfy     $$n^2 = x_1^2 + x_2^2 + x_3^2 + ... + x_k^2 $$ $$x_1, x_2, ...,...
0
votes
1answer
37 views

Composition formula for $ax^2 \pm by^2$

It is well known that $(x^2+y^2)(u^2+v^2)=(xu+yv)^2 + (xv-uy)^2$, thus it suffices to characterize the primes $p$ with $x^2+y^2=p$. Are there any similar composition formulae for $ax^2+by^2$ and $ax^2-...
1
vote
3answers
229 views

The prime power decomposition of sum of two squares

Show that if an integer is a sum of two squares: $n=x^2+y^2$, then in the prime power decomposition of $n$, all primes $p = 3 \pmod{4}$ appear with even exponents : For $p = 3 \pmod{4}$, if $p^k \...
-1
votes
2answers
16k views

Value of: $ \frac {1} {\sqrt {4} + \sqrt {5}} + \frac {1} {\sqrt {5} + \sqrt {6}} + … + \frac {1} {\sqrt {80} + \sqrt {81}}$ [closed]

I have the following question: Find the value of: $$ \frac {1} {\sqrt {4} + \sqrt {5}} + \frac {1} {\sqrt {5} + \sqrt {6}} + ... + \frac {1} {\sqrt {80} + \sqrt {81}}$$ The book ...
1
vote
1answer
125 views

Could be this : $7131372917538397234773191167617941438959$ written as $x^{2}+y^{2}$ with $x, y$ integers? [duplicate]

I have constructed the number $7131372917538397234773191167617941438959$, which is prime as shown here, using all primes under $100$ which contains two digits by randomly ordering them as $31,37,29,\...
0
votes
3answers
191 views

How can I prove the sum of squares and the sum of cubes with binomial coefficients? [closed]

My main problem is starting. I can't "see" anything that might give me an idea to find a relationship between these two things Thank you :)
3
votes
1answer
120 views

Prove or disprove a claim involving Pythagorean primes

As we know, every Pythagorean prime $p$ is expressible as $p=x^2+y^2$ such that $x$ is an odd integer and $y$ is an even integer. Prove or disprove: $x$ is a quadratic residue modulo $p$.
-2
votes
1answer
109 views

Remainder of $\sum_{x=1}^{312} x \times x!$ divided by 2016 [closed]

I have this question: What is the remainder of $$\sum_{x=1}^{312} x \times x!$$ (or just simply) $$(1! \times 1) + (2! \times 2) + (3! \times 3) + \dots + (312! \times312)$$ Divided by ...
0
votes
1answer
97 views

Two times the sum of two squares is a sum of two squares

For all $a,b\in\mathbb Z$ there are $c,d\in\mathbb Z$ such that $2(a^2+b^2)=c^2+d^2$. The conjecture is tested for $a^2+b^2<1,000,000$ but I have problems with proving it.
0
votes
2answers
4k views

Find the value of: $\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$ [closed]

Find the value of:$$\frac{\sqrt{45} + \sqrt{18}} {\sqrt{7+2\sqrt{10}}}$$ I need help solving this question. Every reply is appreciated. Thanks!
0
votes
2answers
84 views

Find the value of $\frac {1} {1^2 + 1} +\frac {1} {2^2 + 2} +\frac {1} {3^2 + 3} +\frac {1} {4^2 + 4} ++ \dots + \frac {1} {2008^2 + 2008} $

I have this question: Find the value of: $$\frac {1} {1^2 + 1} +\frac {1} {2^2 + 2} +\frac {1} {3^2 + 3} +\frac {1} {4^2 + 4} ++ \dots + \frac {1} {2008^2 + 2008} $$ My attempt: I ...
0
votes
0answers
59 views

A method to study the factorization of very big numbers

Is there a method to construct very big numbers in a given interval, with rectangular distribution, by selecting prime factors randomly? I want to study factorizations of the kind $a=b\cdot c$, where ...
-1
votes
3answers
90 views

Is $7m^2-3n^2$ a perfect square?

Is $7m^2-3n^2$ a perfect square for all positive integers $m,n$? I tried using double induction, but failed. Any other approach? By the way, is this related to fermat's theorem on representation of ...
1
vote
0answers
166 views

Sum of two squares of an integer N, the simplest algorithm?

The algorithm is based on the following observation. An integer $b>a$ can be written as $b=a + k$ with a and k integers. So if we want to find a sum of 2 squares of an integer N, we start by ...
0
votes
1answer
209 views

About square free semiprimes which are sums of two squares

This conjecture seems to be true: If the product of two different primes is a sum of two squares, the so are the primes. I've tested it for $pq<1000$, but would like to see a proof. Edit: If $p\...
11
votes
3answers
809 views

Show that $x^2+y^2+z^2=999$ has no integer solutions

The question is asking us to prove that $x^2+y^2+z^2=999$ has no integer solutions. Attempt at a solution: So I've noticed that since 999 is odd, either one of the variables or all three of the ...
3
votes
0answers
107 views

A surprising link between sum of squares and vector product [closed]

When we look at the two smallest numbers $n$ such that $n$ can not be written as the sum of $3$ squares, we get $7$. And we know that there exists a vector product on $\mathbb R^n$ if, and only if $n\...
2
votes
1answer
41 views

Find all prime numbers $y$ and $z$ if $x$ is an integer and $x^2-y^2-z^2=2017.$

Find prime numbers $y$ and $z$ if $x$ is an integer and $x^2-y^2-z^2=2017.$ $x^2-y^2-z^2$ is not factorable, so what should I do? My mind is blank here. Any help is appreciated.
0
votes
1answer
186 views

Algorithm to Divide the number into 4 square part

I am having a number N i want to divide this number into sum of square contains upto 4 numbers. ...
1
vote
2answers
37 views

Summation formula proof

For $k$ and $l$ $\in n$ how can I prove the following : $$(\sum_{k=1}^{k=n}\pi_{k}Y_{k}-\sum_{k=1}^{k=n}Y_{k})^2=\sum_{k=1}^{k=n}\sum_{l=1}^{l=n}Y_{k}Y_{l}(\pi_{k}-1)(\pi_{l}-1)$$ . I have really ...
1
vote
1answer
251 views

Polynomial equal to sum of squares of polynomials [duplicate]

Given a nonnegative polynomial $p(x)$ on $\mathbb{R}$, does there exist some $k$ such that for some polynomials $f_1,\ldots ,f_k$ we have $p(x)=\sum_{i=1}^k(f_i)^2$? I think yes, because of the ...
4
votes
2answers
950 views

What Is The Sum of All of The Real Root

I found this question on my test: What is the sum of all of the real root of $x^3-4x^2+x=-6$? A.) $-4 $ A.) $-2$ A.) $-0 $ A.) $2 $ A.) $4 $ My answer: $...
2
votes
3answers
733 views

Divisibility of the sum of squares of two odd numbers

$A=m^{2}+n^{2}$ with $m$ and $n$ both being odd numbers. We need to find out of this integer is divisible by $2,4$ or $6.$ I don't know how to start solving this. Square of any odd number will have ...
0
votes
0answers
50 views

Given A number N conver it to 2 square form

I have a number N i want to subtract square of x from N , where x is any integer ...