The Prime Glossary: primeval number

primeval number
(another Prime Pages' Glossary entries)

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Have you every played the game where you take a word like "prime" and see how many other words you can find in it? For example, you might find: prime, prim, ripe, imp, pie, rip,... Why not play the same game with integers and look for primes? For example, in 1379 we find the following thirty-one primes:

3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97, 137,
139, 173, 179, 193, 197, 317, 379, 397, 719, 739, 937, 971,
1973, 3719, 3917, 7193, 9137, 9173, and 9371

(Note that we may only use as many of each digit as are present in the original number.)

Mike Keith defined a primeval number to be a positive integer which contains more primes (in the way discussed above) than any smaller positive integer. Below are all of the primeval numbers less than 100,000 (from Keith's web page).

All primeval numbers less than 100,000
primeval integer number of
primes contained primeval integer number of
primes contained

2 1 1379 31

13 3 10079 33

37 4 10123 35

107 5 10136 41

113 7 10139 53

137 11 10237 55

1013 11 10279 60

1037 19 10367 64

1079 21 10379 89

1237 26 12379 96

1367 29 13679 106

Related pages (outside of this work)

Integers Containing Many Embedded Primes by Mike Keith