[QUOTE=CRGreathouse]I grant that there are people who would consider either reasonable. My statement holds for both definitions, and indeed a wider range on both sides.
[/QUOTE]

Sieving efficiency doesn't improve much when one goes deeper than about 10[sup]12[/sup] or so.

[QUOTE=CRGreathouse]If you're interested, go ahead. I'd rather spend the time looking up better estimates of the product (with or without the assumption of the RH).
[/QUOTE]

It was merely a suggestion. I do not intend on doing so.

[QUOTE=3.14159;227196]Well, I sieved to 700M. Apparently, there are still a load of candidates left.[/QUOTE]

[code]ff(x)=1/log(x)/exp(Euler)
s=ff(700e6);forprime(p=2,90,s*=p/(p-1));s[/code]

[QUOTE=3.14159;227198]Sieving efficiency doesn't improve much when one goes deeper than about 10[sup]12[/sup] or so.[/QUOTE]

[code]> ff(1e12)/ff(1e9)
%1 = 0.7500000000000000000000000000[/code]
so going to 1e12 only removes about a quarter of the candidates left at 1e9.

[QUOTE=CRGreathouse]Well, I sieved to 700M. Apparently, there are still a load of candidates left.
[/QUOTE]

I was not making a reference to p(90); I was referring to 1621![sup]2[/sup]

[QUOTE=3.14159;227201]I was not making a reference to p(90); I was referring to 1621![sup]2[/sup][/QUOTE]

In that case,
[code]s=ff(700e6);forprime(p=2,1621,s*=p/(p-1));s
%1 = 0.3648162081214616551560298255[/code]
so about 5/8 of the candidates should be removed by sieving up to 700 million. To get to 2/3 you'd need 5 billion...

Well, I found a PRP somewhat soon: 12066 * 1621![sup]2[/sup] + 1

Record project: k * 15670! + 1 (≈59k digits)

k-range: 100k to 330k.

Odds: 1 in 7880 with no sieving. With sieving to 1.5*10[sup]9[/sup], about 1 in 3900 to 4500, assuming 1/2 of candidates are eliminated.

Sieving should take 5-10 minutes.

This is taking somewhat longer than I thought. I canceled and went for a smaller range after 33 minutes of nothing.

I'm going to settle between 50 and 100 million.

Settled for 315 million, testing: The odds are about 1 in 3890. So I expect a prime in 3890 * 4 minutes, or 1556 minutes, or within 26 hours.

Okay: I sieved for k * 48[sup]7890[/sup] + 1 and k * 1296[sup]5680[/sup] + 1 (Correction, still sieving for the latter. Up to 193 billion)

I have an idea: Make a variant of vk which sieves for primes in em2, where em2 is the set of integers such that n * m + 1 is divisible by a user-specified number.

Let's make a description: em2 gives the liftmod for Mod(-1, x)/(n), where n is any integer, and where x is any integer coprime to n.

em2 gives the set of integers liftmod(Mod(-1, x)/(n)) + a*x.

a will be a user-specified range, and so will the x and n for em2.

It's basically a sieve that searches for prime cofactors of k * n + 1.

Ex: 17228365081076784548444895 * 400[sup]200[/sup] + 1 = 45569 * 455603 * 755617 * 7757609 * p523