... I agree. ... A is defined recursively so, rather than your "depth" (which I'm not sure I understand) I favour a measure "recursion_depth", which would...
In primenumbers@yahoogroups.com, ... Provided that p << N, I guess that the average value of omega(n/p) for the numbers n up to N with least prime divisor p is...
... After the work Mark and Mike did, it really seems that any prime number is Euclid-style accessible. Unfortunately, I have no idea how this could be proved....
... In this case, of course, the tree degenerates into a list, and so is more straightforward to program; and we have simply card(A)=recursion_depth. Defining...
... Somebody has been here before, carried it up to 256 digits, and saved it all in the factordb http://factorization.ath.cx/index.php?id=1100000000024656542 ...
... We only need the smallest factor which is trivially found by trial factoring to be 103. The sequence of smallest prime factors is the Euclid-Mullin...
... It is notable that only one known factorization is incomplete: http://www.rieselprime.de/Others/EuclidMullin.htm and in that case it suffices to show that...
If 0 < b1 < p1 < q and 0 < b2 < p2 < q and b1^p1 = 1 mod q^2 and b2^p2 = 1 mod q^2 What can we conclude about relation between p1 and p2? Consider the case...
... Get Pari-GP to do it for you :-) Assuming that p|q-1, with prime q, we simply ask for bsol(p,q)=lift(znprimroot(q^3)^(q^2*(q-1)/p)); \\ example: ...
... Indeed. Moreover, (b1,q) and (b2,q) are Wieferich pairs with the same q. Such conjunctions are rare. Even when we find one, the probability that it solves ...
... I have proven this for primes p up to 19, but the general proof eludes me. My proof for p = 19 was already laborious and is equivalent to showing that...
... Using 4 days pari-GP at 3.6Ghz I have solidified these results. For b1, b2, q < 10^7.5, the complete list of such q is:- 555383 1767407 2103107 2452757 ...
... Interestingly enough, if p>>1, sum=B_1+log(log(p)) and we get log(log(N))-log(log(p)). The chance of N being prime is 1/log(N). Invoking Mertens' 3rd...
... For log(p) << log(N), the average of omega(n/p), with n running from p to N and p the least prime divisor of n, is (I claim) asymptotic to the sum of 1/q,...
... In the course of this Mike Oakes found 8 pairs of Wieferich pairs not recorded Michael Mosinghoff, who condidered only q = 1 mod 4 when q > 10^7 and hence...
... Also 4232737^17591459 = 1 mod 35182919^2 15865919^17591459 = 1 mod 35182919^2 4144001^19276511 = 1 mod 38553023^2 13038863^19276511 = 1 mod 38553023^2 ...
... Would I be right in supposing, from the facts that (a) these results are appearing in random order of q and (b) very quickly, that you have given this...
... Combination of two effects: several cores and specializing to the most lucrative case, p1=p2=(q-1)/2, so I only look at q if it is the larger member of a...
... These have larger q: 1783447^48649379 = 1 mod 97298759^2 25659449^48649379 = 1 mod 97298759^2 2076259^48687659 = 1 mod 97375319^2 34543973^48687659 = 1 mod...
Prime q=2*p+1 with primes b<c<p such that q^2|b^p-1 and q^2|c^p-1 555383, 1767407, 2103107, 7400567, 12836987, 14668163, 15404867, 16238303, 19572647,...
