Do primes arise from wrong binomic formulae?

When at school, we had to learn formulae called "binomische Formeln" in German; I do not know the English term for them. Search with Google led me to the binomial theorem which is very near, but more general (also for higher powers, not only for quadratic expressions), and which includes the binomial coefficients while the formulae we had to learn had only an explicit 2.

Perhaps you remember the three formulae

          (a + b)² = a² + 2ab + b²
          (a - b)² = a² - 2ab + b²
          (a + b) (a - b) = a² - b²

The difficulty with these formulae was not to use them left to right, but to use them backwards, right to left. As an example, we should simplify a fraction like

          a² - b²                       
          -------  , giving  1/3 (a + b)
          3a - 3b

There seems no connection to prime numbers - or is there a hidden connection? One can test and insert some numbers for a and b. We limit the discussion to positive integers a and b, having a > b, in the following.

Searching for primes, it is not an good idea to start with an expression a² - b²; the mixed formula gives exactly a factorisation, and therefore, the result of a² - b² is not a prime, except for the special case a - b = 1. With this special case, one can reach all odd numbers, and therefore all primes but 2. This is noted on the margin only, here.

Numbers from the first two formulae are not suitable either; they are squares, and a square is not a prime.

It happened now and then that we computed errorneously (a + b)² = a² + b²; I call it error 1. The tentation was to do it like the multiplication rule (the distributive law) which states that (a + b) * 2 = a * 2 + b * 2.

Error 2 is to omit the coefficient 2 of the middle term, i.e. to 'compute' (a + b)² = a² + ab + b². And error 3 is to set the minus sign also with the b², i.e. to 'compute' (a - b)² = a² - ab - b². In the following text, I use this third term with a plus sign for the middle term; this is equivalent: if you can compute a number as a²-ab-b², then you can write it as a²+ab-b² also, for some smaller a and b.

Intresting (at least for me) is that these three error terms

          a² + b²
          a² + ab + b²
          a² + ab - b²

Primes of form a²+b² can be factorised if you work with Gaussian integers. Every prime which gives a remainder of 1 when divided by 4 is of form a²+b², and is shown with yellow background.

For some primes, it happens that we get them by two (or even all three? I do not know) different terms. For instance, one gets 13 both as 3² + 2² = 9 + 4 (error 1 term), and also as 3² + 3 * 1 + 1² (error 2 term).

Primes of form a²+b² have yellow background, not depending on if they allow another "error term" or not. Blue background stands for "only a² + ab + b²", red for "only a² + ab - b²", and green for "both second an third error term is possible".

The table is very economic, maybe it can be called spartanic. Numbers divisible by 2, 3 or 5 are completely omitted. For a range of 30 numbers, there are only 8 possibilities to find a prime. I started the table with these 8 colums. This gives a table which is almost bewildering by its colorfulness.

Then I realized that there is only one color for a column when I double the range of each line to 60 numbers, containing 16 places for primes. That is the format of the table now.

Two of the 16 columns contained only "grey" primes, i.e. primes not of the three simple error terms. Some years later, I tried to get them by further error terms. And indeed: it worked with the three terms 2a² + b², 2a² - b², and a² - 2b². Each term was given a color, and these two columns are now a bit colorful.

Beyond that: all primes got a color, i.e. fit to some error term, no prime is left. In most columns, they all get the same color. Eight columns out of the 16 contain primes of form a²+b² (yellow), two primes of form a²+ab+b² (blue), two colums primes of form a²+ab-b² (red), and two columns contain primes which allow the second and the third "error fromula" (green). The last two columns contain primes for the extra terms (containing the factor 2), and there is a mix of these three colors.

One can even find the primes of form a² + b² in the OEIS (Online Encyclopedia of Integer Sequences), under A002144. Primes of form a²+ab+b² are under A068229, and those of form a²+ab-b² under A141302. And you find also the primes not fitting to any "error formula", under A107169.