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If we know the difference between for instance $5-3=2$ and if we also know the difference between $7-5=2$, can we then predict the difference between $X-7=$? Where $X = 11$.

Is there an equation/algorithm that can predict the difference without knowing $X$? Based on sums or something?

Klangen
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Simon
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  • not really but we can prove that say 5 consecutive differences can't all be the same ( unless they are 5 ( impossible in the odd primes)), in any 5 values at least 2 must share their remainders when dividing by 5. this is a straight application of the pigeonhole principle to primes in arithmetic progressions. –  Jul 27 '17 at 16:38
  • If the Twin Prime Conjecture http://mathworld.wolfram.com/TwinPrimeConjecture.html is false, then we can say there is a number M so that $p_{i+1}-p_i$>2 , where $p_i$ is the $i_th$ prime. – gary Jul 27 '17 at 16:42
  • For any sequence $a_1,a_2,a_3,\cdots$ of which $a_1$ is known, knowing the sequence of differences $a_2-a_1,\ a_3-a_2,\ a_4-a_3,\cdots$ is exactly the same thing as knowing the sequence itself. –  Jul 27 '17 at 16:46
  • we know all primes greater than 3 are 1 or 5 remainder on division by 6 ... –  Jul 27 '17 at 17:08

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