Hypothesis 1: any even number greater than 2 can be decomposed into two prime numbers.
Hypothesis 2: for any number K> 3 there exists at least one R> 0 such that K-R = P and K + R = Q, where P, Q are prime numbers.
From the second hypothesis, the first follows automatically, since if we add both equalities, we get 2 * K = P + Q, that is, an even number on the left.
I drew a graph-polygon, which displays the dependence of K on the number of suitable R for a given number K.
And it turns out that the number of R increases with increasing K.
n = 10^4;
res = {};
ProgressIndicator[Dynamic[k], {4, n}]
For[k = 4, k < n, k++,
c[k] = 0;
For[r = 1, r < k, r++,
If[PrimeQ[k + r] && PrimeQ[k - r],
c[k] = c[k] + 1;
];
];
res = Join[res, {{k, c[k]}}];
];
Graphics[Polygon[res], {Axes -> True}]
Question. How to prove that the count of such R is constantly increasing?
