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A177854
Smallest prime of rank n.
2
2, 3, 11, 131, 1571, 43717, 5032843, 1047774137, 7128418089643
OFFSET
0,1
COMMENTS
The Brillhart-Lehmer-Selfridge algorithm provides a general method for proving the primality of P as long as one can factor P+1 or P-1. Therefore for any prime number, when P+1 or P-1 is completely factored, the primality of any factors of P+1 or P-1 can also be proved by the same algorithm. The shortest recursive primality proving chain depth is called the rank of P (cf. A169818).
LINKS
J. Brillhart, D. H. Lehmer and J. L. Selfridge, New primality criteria and factorizations of 2^m+-1, Math. Compl. 29 (1975) 620-647.
EXAMPLE
The "trivial" prime 2 has rank 0. 3 = 2+1 takes one step to reduce to 2, so 3 has rank 1.
P=131: P+1=132=2^2*3*11. P1[1]=2 has rank 0; P1[2]=3 has rank 1; P1[3]=11: P1[3]+1=12=2^2*3; is one step from 3 and has recursion depth = 2. So P=131 has total maximum recursion depth 2+1 = 3 and therefore has rank 3.
MATHEMATICA
The following program runs through all prime numbers until it finds the first rank 7 prime. (It took about a week.) Fr[n_]:= Module[{nm, np, fm, fp, szm, szp, maxm, maxp, thism, thisp, res, jm, jp}, If[n == 2, res = 0, nm = n - 1; np = n + 1; fm = FactorInteger[nm]; fp = FactorInteger[np]; szm = Length[fm]; szp = Length[fp]; maxm = 0; Do[thism = Fr[fm[[jm]][[1]]]; If[maxm < thism, maxm = thism], {jm, 1, szm}]; maxp = 0; Do[thisp = Fr[fp[[jp]][[1]]]; If[maxp maxp, res = maxp]; res++ ]; res]; i=1; While[p = Prime[i]; s = Fr[p]; [p, s] >>> "prime_rank.out"; s<7, i++ ]
PROG
(PARI) rank(p)=if(p<8, return(p>2)); vecmin(apply(k->vecmax(apply(rank, factor(k)[, 1])), [p-1, p+1]))+1
print1(2); r=0; forprime(p=3, , t=rank(p); if(t>r, r=t; print1(", "p))) \\ Charles R Greathouse IV, Oct 03 2016
CROSSREFS
These are the primes where records occur in A169818.
Cf. A005113, A056637. - Robert G. Wilson v, May 28 2010
Sequence in context: A058114 A042337 A061482 * A273598 A135161 A066100
KEYWORD
hard,nonn,more,nice
AUTHOR
Lei Zhou, May 14 2010
EXTENSIONS
Partially edited by N. J. A. Sloane, May 15 2010, May 28 2010
Definition corrected by Robert Gerbicz, May 28 2010
a(8) from Florian Baur, Sep 05 2024
STATUS
approved