But, thinking about it some more, let me see if I can make a Goldbach type of conjecture of all Integers as sums of three primes with both add and subtract. Now I may have to make one adaption, by saying that 1 is a prime number.
I need to look into whether the Goldbach conjecture holds true for all odd numbers. If it does, I may be able to get away with All Integers are represented as the sum of 2 primes, whereas in Legendre 3 squares are needed.
So let me explore what the Least number I need.
Now the proof of my conjecture above is that the subtraction dismisses the restriction of 4^n(8m+7) and hence the proof. So it is a one paragraph proof.
So, afterall, Goldbach finally delivers some gold nuggets of math.
--- end of what I wrote ---
Before I go to bed, let me get started on this improvement for I sense it is there, ready to be plucked like a fresh plum from the orchard.
I have already proven that All Integers are the sum (with subtraction, or with negative perfect squares) of just 3 perfect squares.
What I hope to now prove is that All Integers are the sum (with subtraction, or with negative primes) of just 2 primes.
Now I need to make a few adaptions in this, in that negative primes are the same as positive primes except with a negative sign.
I also need to include 1 and -1 as primes and that 0 is a prime. That is not too much of a burden for they follow the definition of prime by trivially or by default.
Now in Old Math with their Goldbach focused on odd positive numbers, I see no way of achieving 27.
What two primes sum to 27?
As I go down the list of primes 23, 19, 17, etc. I cannot get a 27. So this tells me that Goldbach conjecture cannot be adapted to the odd positive numbers.
However, if I have all the above of negative primes, and 1, -1, and 0 then I can achieve All Integers.
23= 23 + 0
27 = 29 -2
So let me go to bed with that fix and see if this conjecture allows the primes to achieve all the integers.
I would guess it does since the primes are such a more dense set than ever was the set of perfect squares.
So my conjecture of an improvement of the Goldbach is that All Integers are the sum (subtraction or negative primes added) of just 2 primes.