I know that no number less than 64 bits will fail the MillerRabin tests for all of the first 12 primes. That is, those 12 tests will provide a fully deterministic primality test for all 64 bit numbers. (See http://oeis.org/A014233). I also know that for 32 bit numbers, it suffices to apply the MillerRabin tests for the three bases 2, 7 and 61. (See http://primes.utm.edu/prove/merged.html). Is there a very small list of possible bases (other than the first 12 primes), which will provide a fully deterministic test for 64 bit primes?

$\begingroup$ Is there a particular reason to believe that the first 12 primes are sufficient to cover 64bit primes? From the OEIS list we can only deduce that the 11ish set fails at 61.7 bits; the 12ish set might fail very soon after for all we know $\endgroup$ – DarthGizka Nov 10 '14 at 14:54

$\begingroup$ The OEIS entry explicitly states a(12) > 2^64. $\endgroup$ – Emil Jeřábek supports Monica Nov 12 '14 at 13:38
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According to http://millerrabin.appspot.com/, the 7element set {2, 325, 9375, 28178, 450775, 9780504, 1795265022} works for 64bit integers.

$\begingroup$ Thanks very much  that website (of which I was unaware) answers my question very nicely. I now need to search for some references: I wonder whether Jim Sinclair's 7base solution is optimal? $\endgroup$ – Alasdair McAndrew Jul 12 '12 at 0:49

1$\begingroup$ @Alasdair McAndrew: Surely not. If all you need is 2^64 it should be not only possible but tractable to find a 6base solution. It's not impossible that a 5base solution exists but finding one seems extremely unlikely without advances in the theory. $\endgroup$ – Charles Jul 23 '12 at 17:37

$\begingroup$ How does this even work? I'm seeing failures when trying this set, because the numbers 4033 and 4681 are both pseudoprimes to base 2 and 325. $\endgroup$ – Todd Lehman Jul 30 '14 at 8:50

2$\begingroup$ The conventions used by the author of millerrabin.appspot.com are described in priv.ckp.pl/wizykowski/sprp.pdf , though not particularly clearly. You are supposed to test all bases not divisible by $n$. Note that the definition of base$a$ strong pseudoprimes really only makes good sense for $(n,a)=1$. The commonly presented randomized algorithm, restricting the bases to $1<a<n$, exploits the unrelated fact that any common factor of $n$ and $a$ will make $n$ composite, however this is not literally true for $a\ge n$, one must exclude multiples of $n$. $\endgroup$ – Emil Jeřábek supports Monica Jul 30 '14 at 10:48

3$\begingroup$ Is there a paper or some other verification for Jim Sinclair's 7base set? Google only turned up nothing beyond the nebulous statement on the abovelinked page ("at least 2^64"); was the set tested exhaustively up to 2^641? $\endgroup$ – DarthGizka Nov 10 '14 at 14:52