Outline Mathematics
Number Theory

Primes as differences of squares

Here's a problem to tackle:

  1. Which primes can be expressed as the difference of two squares?

  2. Which primes can be so expressed in two or more different ways?

Solution


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Copyright © 1996-2018 Alexander Bogomolny

Solution

  1. Which primes can be expressed as the difference of two squares?

  2. Which primes can be so expressed in two or more different ways?

Since the gaps between consecutive squares increase 1 3 5 ...,1 3 5 ...,1 2 3 ...,2 4 6 ... it is impossible to express the prime 2 as a difference of two squares,two primes,two squares,two whole numbers. However, any odd prime can be so expressed. Indeed, for any n,

(n + 1)^2 = n^2 + 2n + 1,2n - 1,2n,2n + 1,2n + 3.

If, for a given odd,odd,even prime p, we choose n such that p = 2n + 1, then p will be equal to the difference of two squares, (n + 1)^2 and n^2,(n - 1)^2,n^2,(n + 1)^2. This solves the first part of the problem. For the second part, assume that p is a prime and p = a^2 - b^2. Then

p = (a - b)(a + b),2(a - b),(a - b)(a + b),(a - b)^2,(a + b)^2.

But p being a prime, it has no factors other than 1 and itself. Of the two numbers, a - b and a + b, the second is larger,larger,smaller. Which leads to two equations:

a + b = p and
a - b = 1.

By adding the two equation we see that 2a = p + 1,p + 1,p - 1,2p, or a = (p + 1)/2. Since from the second equation b = a - 1,a + 1,a,a - 1, b = (p - 1)/2,p + 1,p - 1,(p - 1)/2,(p + 1)/2.

It appears that p = a^2 - b^2 defines a and b uniquely. We thus conclude that any odd prime number can be represented as a difference of two squares in a unique way.

References

  1. T. Gardiner, More Mathematical Challenges, Cambridge University Press, 2003, pp. 92-93.

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Copyright © 1996-2018 Alexander Bogomolny

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