- Adding the odd numbers in order produces the square numbers.
- Apart from 2, all primes are odd numbers; the difference between two consecutive squares being odd, every prime can be expressed as the difference between two squares.
- Primes of the form 4n+1 can also be expressed as the sum of two squares (not necessarily consecutive); but primes of the form 4n-1 cannot be expressed as the sum of two squares.
- Form 6n+1, 6n-1 does not follow any such pattern.

Examples: 2 = 1^2 +1^2 : abbreviated here to: 2(1,1)

These are all of form 4n+1 (apart from 2).

- 2(1,1)
- 5(2,1)
- 13(3, 2)
- 17(4,1)
- 29(5, 2)
- 37(6,1) and so on.

3^2 =2^2 -1^2 : here written 3(2,1)

- 3(2,1)
- 7(4,3)
- 11(6,5)
- 19 (10, 9 )

and in general prime p(a,b), where a+b=p and a-b=l.

(Follows from a^2 -b^2 =(a+b)(a-b), where a-b =1.)

Note that this also implies that the squares must be consecutive, otherwise (a-b) is not equal to 1 and a^2-b^2 is a multiple of (a+b), which is not prime.

Other amusements

Primes expressible as the sum of three or more different squares (i.e. without repetition, so 7(2,1,1,1) is not counted.)

- 29(4,3,2)
- 41(6,2,1)
- 53(6,4,1)
- 59(7,3,1)
- 61(6,4,3)
- 71(6,5,3,1)
- 79(7,5,2,1)
- 89(7,6,2)

(No regular pattern here: 59, 71 and 79 are 4n-1; rest are 4n+1)

Other notes:

- 81 = 64+16+1 (or9^2= 8^2 +4^2 +1^2 )
- 100+49 = 81+64+1
- 49 = 36+9+4
- 121 = 81+36+4
- 101 = 31+16+4
- 64+36+1

all 4n+1

1729 = 1728 + 1 = 12^3 +1^3 and 1000 + 729 or 10^3 +9^3

(dozenal 1001 = 1000 + 1 and 1001 = 6E4 + 509)

and in addition to 3^2 +4^2 =5^2 we have 3^3 +4^3 +5^3 =6^3 ; but unfortunately this pattern cannot be extended to the fourth powers...