Alright, what I am going to try to do is walk the reader or student through how a mathematician forms a conjecture and then tests it, and finally looks for, and finds a proof. The proof part is the most difficult.
Let me start the proof. And with my experience of proofs, I find geometry is the best means of a math proof, for the mind is better equipped to deal in geometry than to deal in pushing around numbers of algebra. Algebra is best for calculating people, not proving people. Unless the conjecture is wholly based in algebra, then geometry is of little use.
So how would I make this conjecture be geometrical?
Conjecture: by adding or subtracting two perfect squares, all the primes are yielded.
So the start of this proof for me is to turn it, or translate it into geometry, because the mind, the mind of most humans sees things easiest as a geometry a shape, rather than sees quantity or algebra.
So, I say to myself, how are primes turned into geometry? And I remember the Ancient Greeks had something called "polygonal numbers" such as the perfect-squares made of dots:
. for 1
:: for 4
. . . . . . . . . for 9
. . . . . . . . . . . . . . . . for 16
Now notice something really nice about those geometry perfect squares, that you can add or subtract dots such as for example I subtract 9 from 16 I have remaining:
. . . . . . . and notice it is 7, a prime so that 16-9=7
So here I have geometry and perfect squares and a means of extracting primes from subtraction.
Now notice somthing neat about this L shaped figure of perfect squares when you remove enough of the dots to leave behind a L shaped figure.
For 4-1=3 we have
becomes . . . and the prime 3
Now notice that when you have the L shaped figure from a perfect square that the sum of the L dots is always a odd number and that beyond 2 as prime all the primes are odd numbers.
So that, as we construct all the perfect squares and remove all the dots except for the L shaped skeleton we get many odd numbers.
In fact, we generate All the Odd numbers starting with 3 from the L shaped figure of perfect squares.
So, that gives us a big clue as to where the proof is to be found, that since all Perfect Squares have all the odd numbers nested inside them in the L-shaped figure and the L shaped figure is obtained by subtraction, we need only include addition for a headstart on a proof of the conjecture.