2
$\begingroup$

Given a prime number p, is there always a smaller positive integer exponent k such that p+2^k is also prime? We will call the smallest possible value of this exponent the k*. We could interpret this number k as the position where we could add 1 to the binary expression of p in order to obtain another prime number, the smallest possible one.

For example, 61+2^8 = 317. The prime number 61 has a binary expression of 111101 and its "binary distance" function is equal to 8 (dyb[61]=8). This means that in the 8th position of its binary expression, we need to add a 1 (considering the first position as 0, so we have positions from 0 to 5 in 111101): 100111101, which is equal to the prime number 317. 31+2^4 = 47 dyb[31]=4, since 31 in base 2 is 11111, and 47 in base 2 is 101111.

I made the following implementation:

dyb[n_] := Module[{c = 0, p = 0, i = 1},
  If[n > 2 && PrimeQ[n],
    While[! PrimeQ[n + 2^i], i = i + 1; p = i; c = p; ]; c ]]

Print[dyb[61]]

I would like to obtain a table with a certain quantity of primes, with the first column containing the primes p, the second column containing the binary distance k, and the third column containing the second prime q.

Prime p ----- Binary distance ----- Prime q
  61             8                   317
Improve this question
$\endgroup$
1
  • $\begingroup$ seems like there is a typo at the end of the sentence... We will call the smallest possible value of this exponent the " . perhaps we can call this k* ? please change the edit if this makes no sense $\endgroup$
    – Dunlop
    Commented Jul 23, 2023 at 0:15

2 Answers 2

Reset to default
2
$\begingroup$

Are you looking for something like Grid[Table[p = Prime[n]; {p, dyb[p], p + 2^dyb[p]}, {n, 1, 10}], Frame -> All]?

EDIT: I think this fixes your function:

dyb[n_] := Module[{i = 0}, If[PrimeQ[n], While[! PrimeQ[n + 2^i], i = i + 1;]; i]];
Grid[Table[p = Prime[n]; {p, dyb[p], p + 2^dyb[p]}, {n, 2, 20}], Frame -> All]
Improve this answer
$\endgroup$
3
  • $\begingroup$ It seems that your function dypis not working correctly for all prime numbers, or did I misunderstand something? $\endgroup$
    – Gragarian
    Commented Jul 22, 2023 at 13:26
  • $\begingroup$ In fact. His table demonstrated some primes where it doesn't work. The table should provide p dist q 2 0 3 3 1 5 5 1 7 7 2 11 11 1 13 13 2 17 17 1 19 19 2 23 23 3 31 29 1 31 31 4 47 37 2 41 $\endgroup$
    – Rubens Vilhena Fonseca
    Commented Jul 22, 2023 at 16:14
  • $\begingroup$ @RubensVilhenaFonseca I have edited my answer. $\endgroup$
    – Gragarian
    Commented Jul 22, 2023 at 19:56
1
$\begingroup$ 
dyb[n_] := Module[{i = 0}, If[n > 2 && PrimeQ[n], While[! PrimeQ[n + 2^i], i = i + 1;]; i]]

primes = Select[Range[2, 97], PrimeQ];

resp = Table[dyb[n], {n, primes}];

Grid[Transpose[{primes, resp}], Dividers -> {{2 -> True}, {2 -> True}}, 
    Alignment -> {{Center, Center}}, BaseStyle -> {FontWeight -> "Bold", 
    FontSize -> 14}, FrameStyle -> Directive[Thickness[0.005], 
    GrayLevel[0.7]], Background -> {{GrayLevel[0.95], {LightGray, GrayLevel[0.92]}}}]
Improve this answer
$\endgroup$
1
  • $\begingroup$ Your answer would be better, if you used the code to generate some numerical values. $\endgroup$
    – bbgodfrey
    Commented Jul 23, 2023 at 0:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.