right-truncatable prime73939133 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|
|4||2333 (or 133313)||7||2|
|7||25642 (or 166426)||19||7|
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)
- I. O. Angell and H. J. Godwin, "On truncatable primes," Math. Comp., 31 (1977) 265--267. MR 55:248
- 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.]
- Card, "Patterns in primes," J. Recreational Math., 1 (1968) 93--99.
- J. Walstrom and M. Berg, "Prime primes," Math. Mag., 42 (1969) 232. MR0253974