The aim of this work is to optimize the existing formula based on Wilson's theorem to reduce the magnitude of the computation results. Wilson's theorem states: if p is a prime number, then (p-1)! 1 is divisible by p (p-1)! ≡ -1 (mod p). The function (p-1)! increases very rapidly and reaches huge values. When the values of p are large, the calculations become resource-intensive, so it is necessary to reduce the upper limit of the calculation results.

The purpose of this work is to obtain exact and approximate formulas that calculate the number of partitions of even numbers into sums of pairs of prime numbers.

This is just an attempt to associate sums or differences of prime numbers with points lying on an ellipse or hyperbola. Certain pairs of prime numbers can be represented as radius-distances from the focuses to points lying either on the ellipse or on the hyperbola. The ellipse equation can be written in the following form: |p(k)| |p(t)| = 2n. The hyperbola equation can be written in the following form: ||p(k)| - |p(t)|| = 2n. Here p(k) and p(t) are prime numbers (p(1) = 2, p(2) = 3, p(3) = 5, p(4) = 7,...), k and t are indices of prime numbers, 2n is a given even number, k, t, n ∈ N. If we construct ellipses and hyperbolas based on the above, we get the following: 1) there are only 5 non-intersecting curves (for 2n=4; 2n=6; 2n=8; 2n=10; 2n=16). The remaining ellipses have intersection points. 2) there is only 1 non-intersecting hyperbola (for 2n=2) and 1 non-intersecting vertical line. The remaining hyperbolas have intersection points.