right-truncatable prime

73939133 is the largest prime for which all the initial segments of the decimal expansion are also prime (7, 73, 739, ...). So even if we stop writing before we finish the number, we have still written a prime! Such primes are called right-truncatable primes. (If we allow 1 to be considered a prime, then the largest are 1979339333 and 1979339339.)

These primes are also called by many other (deprecated) names. Card called them snowball primes in 1968 [Card1968]. Michael Stueben named them super-primes after reading Alf van der Poorten's note in 1985 (Math. Int. 7:2 (1985) 40). Walstrom and Berg in 1969 called them prime-primes [WB1969]. Right-truncatable primes for which there is no further possible extension have been called super-prime leaders.

What if we change the base (radix) and again look for right truncatable primes? The following table gives the answer for the first few bases, as well as the tally of right-truncatable primes, and of super-prime leaders.

base	largest right-truncatable prime	tally	leaders
2^*	1011	5	2
3	2122	4	1
4	2333 (or 133313)	7	2
5	34222	14	4
6	2155555	36	11
7	25642 (or 166426)	19	7
8	2117717	68	20
9	3444224222	68	23
10	73939133	83	27

[* Note - strictly there are no base-2 right-truncatable primes, as there are no single-digit (single-bit) primes. However, for this special case, we permit the sequence to terminate at either two (10₂) or three (11₂) instead.]

As the base increases, there are more opportunities for finding extensions from each prime; therefore, heuristically, the number of right-truncatable primes would be expected to increase without bound as the base increases. Similarly, the number of superprime leaders also would be expected to increase without bound. For example, for base 42, there are 175734 primes and 63872 leaders.

Related pages (outside of this work)

RTPs in various bases

References:

AG1977
I. O. Angell and H. J. Godwin, "On truncatable primes," Math. Comp., 31 (1977) 265--267. MR 55:248
Caldwell87
C. Caldwell, "Truncatable primes," J. Recreational Math., 19:1 (1987) 30--33. [A recreational note discussing left truncatable primes, right truncatable primes, and deletable primes.]
Card1968
Card, "Patterns in primes," J. Recreational Math., 1 (1968) 93--99.
WB1969
J. Walstrom and M. Berg, "Prime primes," Math. Mag., 42 (1969) 232. MR0253974