what is the least prime p, such that mod(p, k)=1 for 18 values of k, k<= floor( sqrt(n) ) ? Answer: n=786433 Why would, among the first 100,000 primes, it be so rare for p mod k to occur just 18 times? the count {k , frequency of mod(p,k)=1 } among the first 10^5 primes is {0,2},{1,5380},{2,2897},{3,15798},{4,823},{5,10186},{6,232}, {7,17325},{8,1398},{9,2239},{10,28},{11,12585},{12,9},{13,602}, {14,550},{15,9327},{16,3},{17,2828},{18,1},{19,2043},{20,149}, {21,36},{22,22},{23,6659},{24,42},{25,10},{26,236},{27,422}, {28,0},{29,766},{30,0},{31,2190},{32,13},{33,0}, ... does it mean anything special that 786433 mod k = 1 for k= {2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768} and each of these factors into 2's and 3's? (Pierpont primes A005109) This 786433 also occurs in A070806 (and on the 19th place). Hmmm. Wouter. ------------------------------------ for Mathematica Affectionado's: it2=Table[{Floor[Sqrt[Prime[n]]],Count[Mod[Prime[n],Range@Floor[Sqrt[Prime[n]]]],1]},{n,100000}]; {#,Count[it2,{_,#}]}&/@Range[0,Max[Last/@it2]] ------------------------------------