I suggest the following formula for assigning a weight to a prime

gap discovery:

exp(g / ln(N)) . g . (ln N)^2 . ln(ln N) ..................... (1)

g is the length of the gap, N is one of the endpoints. Or maybe the

midpoint. It doesn't matter too much. The higher the weight, the

more remarkable the discovery.

The formula attempts to take into account both the rarity of the gap

amongst primes of size N, [ exp(g/ln(N)) ] and also the

approximate difficulty (CPU cycles) required to find it/check it.

It may be used to compare PRP gaps with PRP gaps. It may also

be used, if desired, to compare Prime gaps with Prime gaps.

However, for large N, the chief difficulty in proving a prime gap

would be in proving the numbers prime and not just PRP.

The justification for the fomula: Let n = ln(N)

To check the gap itself requires testing o(g) numbers for probable

primality. Each test takes o(n) FFT multiplications. Each FFT

multiplication takes o(n.ln(n)) basic operations. Checking the gap

therefore takes o( g n^2 ln(n) ) CPU cycles.

The rarity of a gap of length g of numbers of this size is of order

exp(-g/n). To actually FIND (not just check) the gap of length g

requires, therefore,

o( g n^2 ln(n) ) / exp(-g/n) = o( exp(g/n) g n^2 ln(n) ) CPU cycles,

for a naive search. Therefore I present the formula (1) above as a

reasonable formula for judging a prime gap or a PRP gap. If people

use nifty brainy tricks to save themselves a few CPU cycles, they

should not be penalised for that. That's why I assume a naive

search to derive the formula.

Note that it CAN NOT be used to compare a prime gap with a PRP

gap. Apples to Oranges, as someone else said.

Yours, Mike H...

Michael Hartley :

Michael.Hartley@...
Head, Department of Information Technology,

Sepang Institute of Technology

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