A Look at Sums of Multiples of Squares

I certainly have a deeper understanding now of what numbers can be expressed as a sum of two squares and in how many ways. See my earlier posts:

• Sums of Square Revisited
• Fundamental Theorem on Sums of Two Squares
• Numbers that are Sums of Two Squares
• Sums of Squares
• Finding Significance in Insignificant Numbers
• Sum of Two Squares
• Sum of Squares of Integers and Catalan Numbers

There's certainly some overlap in these posts but mostly they've dealt with numbers of the form $x^2+y^2$ rather than $a{x}^2+b{y}^2$ where $a$ and $b$ are integers. It's the latter type of expression that I want to explore in this post. My interest was stimulated by today's diurnal age number, 25229, that happens to be a 4k+1 prime.

As a 4k+1 prime, it can be expressed as a sum of two squares in one way only, viz. ${98}^2+{125}^2$. However, it turns out that it can be expressed a sum of two squares multiplied by varying coefficients in many different ways. I created some Sage code to investigate:

number=25229 #enter any number
x,y = var('x'),var('y')
a,b = var('a'), var('b')
for a in range(100):
    for b in range(100):
        for x in range(sqrt(number)):
            for y in range(sqrt(number)):
                if a * x^2 + b * y^2 == number:
                    if a < b:
                        print(a, b, x, y)

Running this code on the SageMathCell Server generated the following results for a <100 and b < 100. The first number represents a, the second b, the third x and the fourth y. For example, the first entry (1, 4, 125, 49) represents ${125}^2+4\times {49}^2$.

(1, 4, 125, 49)
(1, 5, 27, 70)
(1, 7, 41, 58)
(1, 13, 101, 34)
(1, 20, 27, 35)
(1, 25, 98, 25)
(1, 28, 41, 29)
(1, 29, 75, 26)
(1, 49, 125, 14)
(1, 52, 101, 17)
(1, 65, 42, 19)
(1, 85, 152, 5)
(1, 91, 127, 10)
(1, 97, 94, 13)
(2, 3, 61, 77)
(2, 11, 105, 17)
(2, 21, 110, 7)
(2, 51, 37, 21)
(3, 26, 9, 31)
(4, 25, 49, 25)
(4, 65, 21, 19)
(4, 85, 76, 5)
(4, 97, 47, 13)
(5, 6, 71, 2)
(5, 9, 70, 9)
(5, 19, 11, 36)
(5, 24, 71, 1)
(5, 46, 61, 12)
(5, 59, 45, 16)
(5, 69, 8, 19)
(5, 76, 11, 18)
(5, 81, 70, 3)
(5, 89, 69, 4)
(6, 53, 56, 11)
(7, 22, 31, 29)
(7, 29, 60, 1)
(7, 46, 53, 11)
(8, 21, 55, 7)
(9, 20, 9, 35)
(9, 29, 25, 26)
(9, 65, 14, 19)
(11, 18, 17, 35)
(11, 50, 17, 21)
(11, 98, 17, 15)
(13, 29, 36, 17) 

It would be interesting to see the full range of possibilities for a, b, x and y. However, for larger values of a and b, the SageMathCell Server seems to time out. It is easy to see how the case of (1, 4, 125, 49) comes about because: $${125}^2+(2\times 49{)}^2={125}^2+(2\times 49{)}^2={125}^2+4\times {49}^2$$ However, in the case of (2, 3, 61, 77) where $(\sqrt{2}\times 61{)}^2+(\sqrt{3}\times 77{)}^2=25229$, it's far from clear how the result comes about.

It's probably better to start investigating with a smaller 4k+1 prime such as 61. Now $61=5^2+6^2$ and there is only one possibility with coefficients and that is $4\times 2^2+5\times 3^2$. Similarly with $73=3^2+8^2$, there is only one possibility and that is $5\times 3^2+7\times 2^2$. With $97=4^2+9^2$, there are two possibilities: $4\times 2^2+9\times 3^2$ and $5\times 3^2+13\times 2^2$.

It can be noted that 4k+3 primes such as 43, while not able to be represented as a sum of two squares, can be represented as a sum of squares with coefficients: $3\times 3^2+4\times 2^2$. Essentially, the elements of the set of all multiples of the square numbers (let's call the set S) are being combined with themselves by addition to form a set of new numbers (let's call this set N). This set consists of $1^2=1$ and the multiples of 1 are the natural numbers. This of course is trivial because it means for any number n can be written as $(n-m)\times 1^2+m\times 1^2$ where $m<n$.

This result means that it is necessary to exclude 1 from our investigation and begin with $2^2=4$. So all multiples of 4 are permissible, as are all multiples of 9, 25, 36, 49, 64, 81 and so on. In this scheme, the smallest possible number would be $2^2+2\times 2^2=12$ with $2^2+2^2=8$ being ignored since we know the rules governing the requirements for a number to be a sum of two squares. This means that coefficients must not be perfect squares and thus $2^2+4\times 2^2=2^2+4^2=20$ is not acceptable. Even coefficients that add to perfect squares when the squared terms are the same must be excluded.

Thus $2^2+3\times 2^2=2^2\times (1+3)=2^2\times 4=4^2=16$ falls victim. From the outset, it's clear that certain numbers (apart from the numbers 1 to 11) cannot be represented by any combination of multiples of squares: 14, 15, 17, 18, 19, 20, 21 and 23. However, it seems clear that every number above 23 can be represented either as:

• a perfect square e.g. $25=5^2$
• a multiple of a perfect square e.g. $50=2\times 25$
• a sum of two different squares e.g. $25=4^2+3^2$
• a sum of the multiples of two squares e.g. $25=4\times 2^2+3^2$. 

I've not proved this by any means but it seems highly likely, given the increase in possibilities as the numbers get larger. So the main takeaway from all this is that $a{x}^2+b{y}^2=23$ is the last ellipse in the family of ellipses $a{x}^2+b{y}^2=N$, where N is a natural number and a>1 and b>1, that does not have integer solutions for a, b, x and y.