A182380 Primes whose base 26 representation (using a=1, b=2, ..., y=25, z=0) form English words or phrases.

31, 53, 61, 67, 109, 149, 157, 197, 223, 313, 347, 353, 359, 379, 409, 421, 503, 509, 521, 613, 691, 743, 859, 863, 929, 1049, 1097, 1163, 1181, 1201, 1249, 1487, 1489, 1601, 2281, 2437, 2441, 2521, 2579, 2741, 2753

Offset: 1

Author

Patrick Devlin, Apr 27 2012

Keywords

• nonn
• base
• word

Comments

Some particularly pleasing prime words and phrases are (with capitals added merely for visual clarity): [discovered by Patrick Devlin, April 2012]

"somePrime" -> 4092274325963

"somePrimeWordSequence" -> 390521469300124399570501784387

"thisIsAGoodExampleOfAPrimePhrase"

-> 1486423446502142057087542429696717235339605927

And some OEIS-themed prime (pseudo-)words and phrases are:

"NJAS" -> 252869

"integers" -> 76851151747

"welcomeToOEIS" -> 2214931257921335609

"theOEISWordPrime" -> 34075123572372820632427

Let w be any phrase (e.g., w could be Homer's Iliad, or w could be the unabridged concatenation of all of Shakespeare's works). Then Dirichlet's theorem on arithmetic progressions implies that if the last letter of w is relatively prime to 26, then there are infinitely many primes whose final digits base 26 are exactly w. There is no guarantee, however, that these primes would be prime phrases since there is essentially no control over how the beginnings of these base 26 representations would look.

Examples

			The English word "beg" becomes 2*26^2 + 5*26 + 7 = 1489, which is prime, so 1489 is in the sequence.  Similarly, "bee" becomes 1487, which is also prime (thus, "bee" and "beg" are the first 'twin prime words' in this sequence).
		

Links

• C. K. Caldwell, The Prime Lexicon (This is for the related base 36 interpretation as in A038842)

Crossrefs

Cf. A038842 (base 36 version), A072922.

Programs

Maple

# To test if a word  w="someword" [all lowercase]  corresponds to a prime,
# call isprime(wordToNumber(w))  or  ifactor(wordToNumber(w))
letters:=["a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m", "n", "o", "p", "q", "r", "s", "t", "u", "v", "w", "x", "y", "z"]:
wordToNumber:=proc(w) local lastLetter, i:
    if length(w) = 0 then return 0: end if:
    lastLetter := w[length(w)]:
    for i to nops(letters) - 1 do if letters[i] = lastLetter then return i + 26*wordToNumber(w[1 .. length(w) - 1]): fi: od:
    return 26*wordToNumber(w[1 .. length(w) - 1]):
end proc: