What is the smallest number expressible as the sum of 2 different squares in 2 different ways?

Answer 1

Originally, the question did not contain the word 'different', but then it is obvious that the answer is 50 = 5^2+5^2 = 1^2 + 7^2.

/

Why obvious?

/

If we have a square that is itself the sum of squares, say n^2 = a^2+b^2, we can write it as (a+ib)(a-ib), where i is the complex root of -1.

Since 2 = 1^2 + 1^2, it follows that 2 = (1+i)(1-i), so is not prime in the field of complex numbers.

/

We can then write a number of the form 2.n^2 as (1+i)(1-i)(a+bi)(a-bi), and by rewriting it, we arrive at [(1+i)(a+bi)][(1-i)(a-bi)] = [(a-b)+i.(a+b)][(a-b)-i.(a+b) = (a-b)^2 + (a+b)^2, but also as 2.n^2 = n^2 + n^2.

/

Since 25 is the smallest square that can be written as sum of two squares, namely 5^2 = 3^2 + 4^2, we find that 50 = (3-4)^2 + (3+4)^2 = 1^2 + 7^2 on the one hand and 50 = 5^2 + 5^2 on the other hand.

/

The requirement for the squares to be different will make the smallest number a bit larger. We now look for two prime numbers P1 and P2 who are themselves sum of squares. We will see how that works. It is well known from algebra that it is sufficient for primes to be 1 mod 4, in order for them to be the sum of two squares. The smallest of these primes are 5 and 13. Note that 5 = 1^2 + 2^2 and 13 = 2^2 + 3^2.

/

Their product 65 = 5.13 is the smallest number that can be written as the sum of two different squares in two different ways. This is seen by writing out the complex multiplications and changing the order of the factors, as follows:

/

65 = 5.13 = [(1+2i)(1-2i)].[(2+3i)(2-3i)]

= [(1+2i)(2+3i)].[(1-2i)(2-3i)] = [(2-6)+i.(3+4)][(2-6)-i.(3+4)] = 4^2 + 7^2.

= [(1+2i)(2-3i)].[(1-2i)(2+3i)] = [(2+6)+i.(4-3)][(2+6)-i.(4-3)] = 8^2 + 1^2.

/

This method can be used to find numbers that are smallest expressible as the sum of 2 different squares in N (N>2) ways.

We need the next few primes 1 mod 4, which are 17, 29 and so on, and use either these or their even powers (squares, squares of squares, cubes of squares, etc.) The reader may try and find a different criterion for N even or odd.

/

For N = 3, we find

325 = 5.5.13

= 1^2 + 18^2

= 6^2 + 17^2

= 10^2 + 15^2.

/

For N = 4 we find

1105 = 5.13.17

= 4^2 + 33^2

= 9^2 + 32^2

= 12^2 + 31^2

= 23^2 + 24^2.

/

For N = 5 we find

8125 = 5.5.5.5.13

= 5^2 + 90^2

= 27^2 + 86^2

= 30^2 + 85^2

= 50^2 + 75^2

= 58^2 + 69^2

/

These numbers tend to get bigger, but alas, the next in the sequence is smaller.

/

For N = 6 we find

5525 = 5.5.13.17

= 7^2 + 74^2

= 14^2 + 73^2

= 22^2 + 71^2

= 25^2 + 70^2

= 41^2 + 62^2

= 50^2 + 55^2

/

And so on.

The reader may verify these results.