For any positive integer n>1, define gap(n) as(adsbygoogle = window.adsbygoogle || []).push({});

gap(n) = |n-p| = |n-q| for p,q the two primes closest to n such that p+q = 2n.

The Goldbach Conjecture is equivalent to the existence of p and q, and thus the existence of gap(n), for all n>1. (Or all n>2 if p and q are required to be odd primes, which does not affect the substance of the conjecture.)

But I make the following conjecture, which is stronger than the mere existence of gap(n) for all n>1:

For all n>1, the maximum value of gap(n)/n is gap(22)/22 = 9/22 = .40909... (for primes p=13 and q=31).

Other high values of gap(n)/n are

gap(8)/8 = 3/8 = .375 (p=5, q=11)

gap(46)/46 = 15/46 = .326... (p=31, q=61)

gap(28)/28 = 9/28 = .321... (p=19, q=37)

gap(32)/32 = 9/32 = .28125 (p=23, q=41)

gap(58)/58 = 15/58 = .259... (p=43, q=73)

gap(4)/4 = 1/4 = .25 (p=3, q=5)

Other n for which gap(n)/n > .2 include [in descending order of gap(n)/n] n=49, 25, 38, 146 [gap(146)=33 for p=113, q=179], 9, 68, 55, 14, 24, 74.

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# Stronger Goldbach-related conjecture

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