Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [sums-of-squares]

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

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

These topics include, for example:

737 questions
7
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3 answers

How to find the sum of squares of $1\ldots N-1$ that add to a squared number $N^2$?

So here's a question my friend recently gave me and ever since I've been trying to solve it without much success: There's a number $N$, and out of the set $U = \{1,2,3,\ldots,(N-1)\}$ we have to find a subset $S$ such that the sum $\sum_{i\in S} i^2…
Weezy
  • 61
3
votes
0 answers

How to prove Sum of squares of $n$ numbers is unique

I was solving a programming problem where I needed to prove that sum of squares of $n$ numbers is unique. i. e. if calculate the sum of squares of equal number of positive integers then if they are same it means that all numbers are the same. How do…
tapananand
  • 131
3
votes
1 answer

Is there a pattern defining the existence of root integer distances in an isometric grid?

In a standard square grid pattern the distances to integer root locations is simply the sum of two squares. We find that these distances have $\sqrt1, \sqrt2, \sqrt4, \sqrt5, \sqrt8, \sqrt9, \sqrt10, \sqrt13, ...$ But, knowing it's the sum of two…
Tatarize
  • 177
3
votes
1 answer

Upper bound on sum of square of integers

I have $n$ non-negative integers $x_1, \dotsc, x_n$ which satisfy the constraint $\sum x_i = S$ I want to derive a bound on $\sum x_i^2$. An easy bound can be calculated as: $\sum x_i^2 \le (max_{x_i}) \sum x_i = S^2$ This bound works for…
btan
  • 133
2
votes
5 answers

Formula for finding sum of three squares that equal a fourth square

I am investigating the sum of three squares that equal a fourth square value. I am only interested in positive integers greater than zero. Specifically i am collecting formulae that identify values of these combinations. Of course i have the…
Hector
  • 219
2
votes
2 answers

Difference of two sums of three squares

I have proved that every integer is the difference of two sums of three squares, i.e., $n = (a^2 + b^2 + c^2) - (d^2 + e^2 + f^2)$ Is this result publishable?
Lateef Suaib
  • 41
2
votes
2 answers

Finding $\sqrt{(14+6\sqrt 5)^3}+\sqrt{(14-6\sqrt 5)^3}$

Find $$\sqrt{(14+6\sqrt{5})^3}+ \sqrt{(14-6\sqrt{5})^3}$$ A.$72$ B.$144$ C.$64\sqrt{5}$ D.$32\sqrt{5}$ How to cancel out the square root?
Ray Jasson
  • 425
1
vote
4 answers

Represent an integer as a sum of n non-consecutive squares

Is it possible to decompose an integer into a sum of n unique squares ,even though they are not necessarily consecutive. For instance, How would I obtain the sequence 1*1 + 7*7 + 14*14 + 37*37 given the integer 1615 or 11*11 + 15*15 + 29*29 + 43*43…
user1476580
1
vote
0 answers

Sums of four squares with fewer squares

Lagrange's four-square theorem states that every natural number can be represented as the sum of four integer squares. $$p=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}$$ where the four numbers $a_{0},a_{1},a_{2},a_{3}$ are integers (source). I would like…
PeterMacGonagan
  • 11
1
vote
0 answers

Finding the pdf of sum of squared weighted gaussian variables

I have 3 sets of weighted gaussians that are part of 3 different Gaussian Mixture Models, phi1,phi2 and phi3. phi1 has n1 gaussian components with weights wj, mean mu_j and variance eta_j_squared, j ranging from 1 to n1. phi2 has n2 such components…
SometimesConfused
  • 11
1
vote
3 answers

Sum of square roots...........

$$ \mbox{If}\quad S = 1 + \,\sqrt{\,\frac{1}{2}\,}\, + \,\sqrt{\,\frac{1}{3}\,}\, + \,\sqrt{\,\frac{1}{4}\,}\, + \,\sqrt{\,\frac{1}{5}\,}\, + \cdots + \,\sqrt{\,\frac{1}{100}\,}\,\,, $$ then what is the value of…
crimsonKnight
  • 162
1
vote
5 answers

$5^m$, where m is any natural, can be expressed as the sum of two perfect squares?

Prove that for all natural $m$, $5^m$ can be expressed as the sum of two perfect squares. Also, prove that $5^m + 2$ can be expressed as the sum of three perfect squares.
Ronald Becerra
  • 865
1
vote
2 answers

Why is there a pattern for making orders of perfect squares (first one, second one, third one) by simply adding two to the next adding each time?

For example, if I had a perfect square of $16$, which is the fourth perfect square, I would add nine to get to the fifth perfect square, $25$. This is probably how it goes:$$1+3=4+5=9+7=16+9=25+11=36+13=49+15=64+17=81+19=100$$and it goes on…
VladTheSmart
  • 33
0
votes
1 answer

how can i calculate square sum of all the products of the summation?

Rule.  The square of a sum is equal to the sum of the squares of all the summands plus the sum of all the double products of the summands in twos: But I do not know how to calculate the square of summation of products >>
bebo
  • 3
0
votes
0 answers

Does Bachet's conjecture hold if the numbers that are squared are required to be unique?

Some backstory, forgive me for not having any kind of mathematical background: A friend of mine received an optional programming assignment to write an algorithm to find the four squares that sum to a given 5-digit number. The professor asserted…
J.A.K.
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