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Call a number 
a prime string from the left if and all numbers obtained by successively removing the rightmost
Digit are Prime. There are 83 left prime strings in base 10. The first few are 2, 3, 5, 7, 23, 29, 31, 37, 53, 59,
71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, ... (Sloane's A024770), the largest being 73,939,133. Similarly,
call a number a prime string from the right if and all numbers obtained by successively removing the left-most
Digit are Prime. The first few are 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 137, 167,
173, ... (Sloane's A033664). A large right prime string is 933,739,397.
See also Prime Array, Prime Number
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.
Rivera, C. ``Problems & Puzzles (Puzzles): Prime Strings.''
http://www.sci.net.mx/~crivera/puzzles/puzz_002.htm.
Sloane, N. J. A. Sequence
A024770
in ``The On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html.033664