Ytukyg

Denote $p[i]$ as the $i$th prime. In my opinion, the following is true:

Prime Gap Axiom
There are always distinct prime factors for ${p[i],p[i]+1,p[i]+2, dots , p[i+1]}$.

Question 1
How to give some good algorithm to dispatch distinct prime factors?

Corollary 1

$p[i+1]-p[i]le 1+pi( p[i+1]/2 ) le i$, for any integer $ige 1$;
There exists at least one prime in the interval $( n,n+pi(n)]$.

Corollary 2
Define Crystal Number Sequence, Crystal(2,3)={2^i* 3^j|i>=0,j>=0}, then the interval ( Crystal[n],Crystal[n+2]) contains at least one prime, for any $nge 1$.

Clearly, Crystal Numbers are very related with Mersenne numbers.

indexTop = 30;

Crystal[p1_, p2_] := 

 Sort@Flatten@Table[p1^i*p2^j, {i, 0, indexTop}, {j, 0, indexTop}]

 list = Take[Crystal[2, 3], indexTop]

Graphics[Point[{#, 0}] & /@ list,

 Epilog -> {Orange, Point[Table[{Prime[k], 0}, {k, 1, indexTop}]]}]

Question 2:
Can we find (very large) three close numbers c1,c2,c3 which are alike $2^i*3^j$？
exclude composite numbers of the interval (c1, c3) , if there is only one survivor, it must be prime! Is this method feasible? will it be much greater than $2^{57885161} − 1$ ?

Link to Mathematica Notebook

Today, I had found an iterative algorithm to dispatch prime factors. Suppose we has the proper primeFactorsList for {1,2,3,...,n-1}, now considering n, n has PrimeNu(n) prime factors, we choose the one that is not dispatched most recently!

(*Dispatch Prime Factors for n<128 *)



reversePrimeFactorsList = {3, 2, 1};

n = 3;

i = 2;

PrimeFactors[x_] := FactorInteger[x][[All, 1]];



While[n < 128,

  n++; If[PrimeQ[n], i++; PrependTo [reversePrimeFactorsList, n],



   pos = Max[ 

            FirstPosition[reversePrimeFactorsList, #] & /@ PrimeFactors[n]

            ];

    PrependTo[reversePrimeFactorsList, reversePrimeFactorsList[[pos]]]

   (*choose the one that is not used most recently*)

   ]

  ];

{i, Prime[i], n}

reversePrimeFactorsList

ListPlot[{Range[n], Reverse@reversePrimeFactorsList}, Filling -> Axis]

Out34:= {31, 127, 128}

Out35:= {2, 127, 3, 5, 31, 41, 61, 11, 5, 17, 59, 13, 29, 23, 19, 113, 7, 37,
11, 109, 2, 107, 53, 3, 2, 103, 17, 101, 5, 11, 7, 97, 3, 19, 47, 31,
23, 13, 2, 89, 11, 29, 43, 17, 7, 83, 41, 3, 5, 79, 2, 11, 19, 5, 37,
73, 3, 71, 7, 23, 17, 67, 11, 13, 2, 3, 31, 61, 5, 59, 29, 19, 7, 11,
3, 53, 13, 17, 2, 7, 3, 47, 23, 5, 11, 43, 7, 41, 2, 13, 19, 37, 3,
5, 17, 11, 2, 31, 2, 29, 7, 3, 13, 5, 2, 23, 11, 7, 5, 19, 3, 17, 2,
3, 7, 13, 2, 11, 5, 3, 2, 7, 3, 5, 2, 3, 2, 1}