A261458 Primes which remain prime after one, two, three and four applications of the rotate-and-add operation of A086002.

10010905789, 10028905771, 10036905763, 10050905749, 10056905743, 10060905739, 10070905729, 10080905719, 10092905707, 10098905701, 10102905697, 10106905693, 10108905691, 10112905687, 10130905669, 10160905639, 10172905627, 10176905623, 10188905611, 10190905609

Offset: 1

Author

Chai Wah Wu, Aug 20 2015

Keywords

• nonn
• base

Comments

There are no primes that remain prime each time after 1,2,...,6 rotate-and-add operations. Proof: by the comment in A086004, such a prime p must have an odd number of digits and must remain so after 1,2,...,5 rotate-and-add operations. Let p have 2m+1 digits, and denote the first and the (m+2)-th digits as (a,b). After a rotate-and-add operation, these digits become (c,d). It is clear that c >= a+b, d >= a+b, except when a carry occur at these digits. If a carry occurred at the (m+2)-th digit, then a carry occurred at the first digit as well. In any case when a carry occurred at these digits, the number of digits is increased by 1 and thus will have even number of digits. This implies that for such a prime p, a carry did not occur after each of the 5 rotate-and-adds. The best one can do is if (a,b) = (1,0), after 4 rotate-and-adds the digits becomes (1,1), (2,2), (4,4), (8,8) or larger and thus a carry will have occurred after at most 5 rotate-and-adds, so such a prime does not exist. - Chai Wah Wu, Aug 21 2015

Examples

			Applying rotate-and-add to the prime 10010905789 four times results in 15800815799, 31600631599, 63200263199, 126399526399, all of which are prime.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..542

Crossrefs

Cf. A086002, A086003, A086004.