OFFSET
1,1
COMMENTS
Let V = (b(1), b(2), ..., b(k)), where k > 1 and b(i) are distinct integers > 1 for j = 1..k. Call p a V-prime if the digits of p in base b(1) spell a prime in each of the bases b(2), ..., b(k). Call p a strong V-prime if p is a (b(j), ..., b(k))-prime for each of the tuples (b(j), ..., b(k)), for j = 1..k-1.
a(157) > 10^11. - Hiroaki Yamanouchi, Oct 25 2015
LINKS
Hiroaki Yamanouchi, Table of n, a(n) for n = 1..156
EXAMPLE
Let p = 171472673. Confirmation that p is a strong (2,3,5,7)-prime follows.
Base-2 for p: u = (1,0,1,0,0,0,1,1,1,0,0,0,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0,1);
u in base 3 spells the prime 8488002487771;
u in base 5 spells the prime 7749195106457425001;
u is base 7 spells the prime 67054080721013093290423.
Base-3 for p: v = (1, 0, 2, 2, 2, 1, 1, 2, 2, 2, 0, 1, 0, 2, 1, 2, 0, 2);
v in base 5 spells the prime 838940251427;
v in base 7 spells the prime 243692337097757.
Base-5 for p: w = (3, 2, 2, 3, 4, 4, 1, 1, 1, 1, 4, 3);
w in base 7 spells the prime 6598716743.
MATHEMATICA
{b1, b2, b3, b4} = {2, 3, 5, 7}; z = 10000000;
Select[Prime[Range[z]],
PrimeQ[FromDigits[IntegerDigits[#, b1], b2]] &&
PrimeQ[FromDigits[IntegerDigits[#, b1], b3]] &&
PrimeQ[FromDigits[IntegerDigits[#, b1], b4]] &&
PrimeQ[FromDigits[IntegerDigits[#, b2], b3]] &&
PrimeQ[FromDigits[IntegerDigits[#, b2], b4]] &&
PrimeQ[FromDigits[IntegerDigits[#, b3], b4]] &]
(* Peter J. C. Moses, Sep 27 2015 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Clark Kimberling, Oct 03 2015
EXTENSIONS
a(4)-a(22) from Hiroaki Yamanouchi, Oct 25 2015
STATUS
approved