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  • A triplet of primes: (pa, p, pb) is such that d = p - pa = pb - p d > 0 Example: (11, 17, 23) d = 6 Now (p - d)*(p + d) = pa * pb Or: p^2 - d^2 = pa * pb And: p^2 = pa * pb + d^2 Represents a right triangle with: hypotenuse = p leg = d leg = (pa * pb)^0.5 It can be demonstrated that there is no other triplet of primes (pa', p', pb') with d' = p' - pa' = pb' - p' and p ≠ p' such...
    Leonel Morales Mar 21
  • Out of curiosity, how did you find the five TPMs? I am trying to find two pairs of prime pairs (pa, pb) and (pa', pb') all greater than 2, such that: for n = (pb - pa)/2 // pb > pa so n > 0 and equivalent n' calculated in the same way n and n' are integers and d = pb - n = n - pa // d integer, d' calculated in the same way d/n = d'/n' or pa/n = pa'/n' and pb/n = pb'/n' I am...
    Leonel Morales Mar 21
  • Wouldn' t it be enough if given prime pn the formula calculates pn+1? All tests for primality would have to be reviewed. Any algorithm depending on primality testing could get a boost in performance from the formula. Cryptography and on line security could be severely affected. Knowing that no prime is missing in the sequence someone could be tempted to create a new encoding system...
    Leonel Morales Apr 3, 2013
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  • Thanks a lot Alan! I really appreciate you explanation. And then when 3 is involved d = 6k +- 2. I have calculated thousands of those and haven't noticed that. I have looked at the sequences of values of d: for a single prime p all the possible values of d, and for the set of primes, all the minimum values of d (http://oeis.org/A078611), all the maximum values, and several others...
    Leonel Morales Feb 10, 2013
  • Quite an interesting problem in fact, full of features and properties. I have been studying Goldbach partitions for even numbers of the form 2p where p is prime, that leads exactly to what Bob points. I have computed thousands of this partitions forming triplets of primes (pa, p, pb) where pa + pb = 2p, p at the center. For example, for p = 43 the triplets would be: (19, 43, 67...
    Leonel Morales Feb 10, 2013