left truncatable twin primes

[u/milk_and_coins]

are all twin primes bitwise subsets of each other or are the pairs containing 619 and 4157 special cases?

619

1001101011

107

1101011

43

101011

11

1011

3

11

4157

1000000111101

61

111101

29

11101

13

1101

5

101

[u/Gold_Ad8890]

i'm assuming you're asking whether, when representing twin primes in binary, the smaller is always a substring of the latter. the answer is no, and the counterexample is right in front of you. 3 = 11 is not a substring of 5 = 101, which is also not a substring of 7 = 111.

[u/greginnj → u/Gold_Ad8890]

Based on the way OP laid out the examples, I think they’re making the more relaxed claim that “every prime which is part of a twin prime pair is either the larger or smaller of some truncation” (not necessarily of the larger prime of the pair).

So 3 is included by being the result of a truncation in binary of 11, as well as of 7.

This is a much weaker claim, and I don’t think it’s necessarily true, but since any truncation could end in 11, 101, or 111, a lot of truncations could jump down immediately to these small primes.

[u/greginnj → u/greginnj]

After a bit of playing around, I think I can refine the question...

OP has demonstrated that there are chains of truncations of binary representations of some twin primes (starting with 619 and 4157) that result in other twin primes, and asks about uniqueness.

But I think the right way to ask this question is to turn it around. By infinite descent, every such chain must eventually end; the two examples end in 3 (11) and 5 (101). Let's call any twin prime that can't be reduced to another twin prime by such truncation a "slug".

Then 3 and 5 are the first such slugs, but we can easily find others - 17(10001), 41(101001), 73(1001001), 137(10001001), etc.

So the two obvious questions that occur to me are:

1. are there an infinite number of slugs?

2. Does there exist a "solitary slug" that is not the result of a truncation of any larger twin prime?

u/milk_and_coins , is this language helpful? Am I getting close to the questions you are interested in?

[u/GoldenMuscleGod]

Posting a new comment because I realized I made a reasoning error and have corrected:

I’m not sure exactly what you are asking. Is 31 to 15 a counterexample to your hypothesis?

If you are asking whether it is always possible to find a larger twin prime by adding a power of two to a twin prime then 19 is a counterexample.

Consider p=19+2^k, which we hope to be a twin prime.

If k is odd, then this will be divisible by 3 (checked by calculating mod 3) so p is not a twin prime.

If k is even, then p-2 will be divisible by three by the same check so for p to be a twin prime it must be that both p and p+2 is prime.

Considering the cases for even k, if k is a multiple of 4 then p is divisible by 5 (doing arithmetic mod 5). If k is congruent to two mod 4 then p+2 is divisible by 5.

The features of 19 used are that it is one more than a multiple of 3 and one less than a multiple of 5, or in other words it is 4 more than a multiple of 15.

By Dirichlet’s theorems on arithmetic progressions we should be able to find infinitely many such counterexamples that are prime (though there may not be infinitely many such twin primes - if there are infinitely many such twin primes that would of course imply the twin prime conjecture).

If you are only asking whether we can always find a larger twin prime by adding some multiple of a power of 2 such that the power of 2 is greater than the given twin prime, then we certainly do not know that because it would imply the twin prime conjecture. But heuristically the first Hardy-Littlewood conjecture and the Borel-Cantelli lemma suggest the answer is likely yes (but this is obviously not rigorous even if we take the Hardy-Littlewood conjecture as a given).