A293142 Largest nonrepunit base-n circular prime (conjectured).

7, 1013, 3121, 211

Offset: 3

Author

Felix Fröhlich, Oct 01 2017

Keywords

• nonn
• base
• more

Comments

A circular prime is a prime where all numbers produced by cyclic permutations of the digits are also prime.

No nonrepunit circular prime exists in base 2, since any nonrepunit prime contains at least one 0 digit in its base-2 representation that yields an even number and thus a composite when permuted to the least significant place, so the offset of the sequence is 3.

a(3)-a(6) were found via a brute-force approach searching from the largest prime with 12 base-n digits backwards. The number of base-n digits in a(n) for n = 3, 4, 5, 6 is 2, 5, 5, 3, respectively. Since this is much shorter than 12 digits, it is conjectured that the terms are the maximal circular primes for those bases. This also verifies that no circular primes with a length between A055642(a(n)) and 13 digits exist in bases 3, 4, 5 and 6.

Candidates for a(7), a(8) and a(9) are 13143449029, 16244441 and 4717103, respectively.

a(10) is probably 999331. If not, it must have more than 23 digits (cf. De Geest link).

Examples

			1013 written in base 4 is 33311. The base-4 numbers 33311, 33113, 31133, 11333, 13331 written in base 10 are 1013, 983, 863, 383 and 509, respectively. All those base-10 numbers are prime and since there is no larger prime up to 12 base-4 digits where all cyclic permutations of base-4 digits are primes, 1013 is conjectured to be the largest nonrepunit circular prime in base 4.
		

Links

• P. De Geest, Circular Primes, World!Of Numbers.

Crossrefs

Cf. A016114, A068652.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293658 (b=5), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293662 (b=9), A293663 (b=10).

Programs

PARI

rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is_circularprime(p, base) = my(db=digits(p, base), r=rot(db), i=0); if(vecmin(db)==0, return(0), while(1, dec=decimal(r, base); if(!ispseudoprime(dec), return(0)); r=rot(r); if(r==db, return(1))))
a(base, maxlength) = my(p=precprime(base^maxlength)); while(p > 2, if(vecmin(digits(p, base))!=vecmax(digits(p, base)), if(is_circularprime(p, base), return(p))); p=precprime(p-1))
for(n=3, 6, print1(a(n, 12), ", ")) \\ start searching a(n) from largest prime with 12 base-n digits backwards