David Wells’ book on Prime Numbers notes the parallels between lucky numbers and primes, suggesting the role of the sieve (p147/8 under “random” primes). Imagine what the effect would be if one prime was no longer considered to be prime. When one realises the interdependence of the primes some of the relationships can be seen:
The pattern in the primes changes at each p squared, but the distance between each p squared is not independent – it is determined by smaller prime numbers: thus the pattern of the primes is fixed between 9 and 16; the length of this pattern can be determined by the simple formula 2.PF.a+a squared where PF is the square root of the lower boundary (a prime) and a is the difference between this prime and the next smallest prime (i.e. pn – pn-1). Thus if we look at each largest gap in the prime numbers we find that we can predict a relatively long section of prime numbers with the same pattern i.e. the gap between 113 and 127 of 14 causes the pattern between 12,769 and 16,129 to be uninterrupted. Of course noone has ever noticed this as the pattern starting at 12,769 is a mere 113# long (i.e. big!) so the run up to 16,129 presents just a fraction of the pattern before the new pattern starts at 16,129.