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For which primes P is the following true?

  1. Aug 20, 2012 #1
    For which primes "P" is the following true?

    the function f(x) = x(x - 1) + p gives you a prime number for all x < p
    I've tried this with 5,11, and 41, but it doesn't work for 7 since 5(5-1) + 7 is not a prime.
    Btw, this isn't homework or anything, just a curiosity.
     
  2. jcsd
  3. Aug 20, 2012 #2
    Re: For which primes "P" is the following true?

    See Euler's Lucky Numbers http://oeis.org/A014556
     
  4. Aug 20, 2012 #3
    Re: For which primes "P" is the following true?

    I guess my question is, why does it work for the numbers that it works for? Is there a way to know whether it will work for a certain prime without checking for every value below it? Are the numbers in that link the only "Lucky Numbers?"
     
  5. Aug 20, 2012 #4
    Re: For which primes "P" is the following true?

    There is a link within the page for a more complete listing of "Lucky Numbers"; however, I doubt that it is a complete listing very much. As to why certain primes are or are not lucky numbers, there is not much to say other than the fact that lucky numbers greater than 3 must be primes of the form 6*n -1 since otherwise 3|(1*2 + P). That being noted, I doubt that there is a simple test other than checking all non-negative numbers < P. You will find that many properties of primes are not easily explained in a simple manner.
     
    Last edited: Aug 20, 2012
  6. Aug 21, 2012 #5
    Re: For which primes "P" is the following true?

    The prime p has to be the smaller of a pair of twin primes. That thins out the candidates considerably.
     
  7. Aug 21, 2012 #6
    Re: For which primes "P" is the following true?

    This is only necessary but not sufficient: for example take p=29 and x = 3: 29 + 6 = 35 not prime

    More unseful information at http://mathworld.wolfram.com/LuckyNumberofEuler.html
     
  8. Aug 21, 2012 #7
    Re: For which primes "P" is the following true?

    All twin primes are of the form 6n - 1 and 6n + 1. The smaller is always 6n - 1, which is what ramsey said.
     
  9. Aug 22, 2012 #8
    Re: For which primes "P" is the following true?

    With ARIBAS it is very simple to get a function 'IsLuckyNumber(n)', returning
    the number of primes in f(i) = i(i - 1) + n with i a natural number < n
    Code (Text):

    ==> IsLuckyNumber(41).
    IsLuckyNumber: i = 1 ; is prime 41
    IsLuckyNumber: i = 2 ; is prime 43
    IsLuckyNumber: i = 3 ; is prime 47
    IsLuckyNumber: i = 4 ; is prime 53
    IsLuckyNumber: i = 5 ; is prime 61
    IsLuckyNumber: i = 6 ; is prime 71
    IsLuckyNumber: i = 7 ; is prime 83
    IsLuckyNumber: i = 8 ; is prime 97
    IsLuckyNumber: i = 9 ; is prime 113
    IsLuckyNumber: i = 10 ; is prime 131
    IsLuckyNumber: i = 11 ; is prime 151
    IsLuckyNumber: i = 12 ; is prime 173
    IsLuckyNumber: i = 13 ; is prime 197
    IsLuckyNumber: i = 14 ; is prime 223
    IsLuckyNumber: i = 15 ; is prime 251
    IsLuckyNumber: i = 16 ; is prime 281
    IsLuckyNumber: i = 17 ; is prime 313
    IsLuckyNumber: i = 18 ; is prime 347
    IsLuckyNumber: i = 19 ; is prime 383
    IsLuckyNumber: i = 20 ; is prime 421
    IsLuckyNumber: i = 21 ; is prime 461
    IsLuckyNumber: i = 22 ; is prime 503
    IsLuckyNumber: i = 23 ; is prime 547
    IsLuckyNumber: i = 24 ; is prime 593
    IsLuckyNumber: i = 25 ; is prime 641
    IsLuckyNumber: i = 26 ; is prime 691
    IsLuckyNumber: i = 27 ; is prime 743
    IsLuckyNumber: i = 28 ; is prime 797
    IsLuckyNumber: i = 29 ; is prime 853
    IsLuckyNumber: i = 30 ; is prime 911
    IsLuckyNumber: i = 31 ; is prime 971
    IsLuckyNumber: i = 32 ; is prime 1033
    IsLuckyNumber: i = 33 ; is prime 1097
    IsLuckyNumber: i = 34 ; is prime 1163
    IsLuckyNumber: i = 35 ; is prime 1231
    IsLuckyNumber: i = 36 ; is prime 1301
    IsLuckyNumber: i = 37 ; is prime 1373
    IsLuckyNumber: i = 38 ; is prime 1447
    IsLuckyNumber: i = 39 ; is prime 1523
    IsLuckyNumber: i = 40 ; is prime 1601
    -: 40

    ==>
     
     
  10. Aug 22, 2012 #9
    Re: For which primes "P" is the following true?

    3 and 5? 3 satisfies the OP's condition. For that matter so does 2.
     
  11. Aug 22, 2012 #10
    Re: For which primes "P" is the following true?

    FYI, Euler's Lucky Numbers map to the last 6 Heegner #s by the rule 4p - 1. So, a good place to begin to learn more about them is by learning more about the Heegner #'s.

    http://en.wikipedia.org/wiki/Heegner_number
     
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