Problems & Puzzles: Puzzles

Puzzle 1221 A puzle about twin primes Long time ago G..L. Honaker, Jr. sent the followin curio & puzzle: 59# + 61 is prime (1922760350154212639131). Note that {59, 61} is a twin prime pair. Q. Find a all the examples alike you can find

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From May 17 to 23, 2025, contributions came from Michael Branicky, Paul Cleary, Oscar Volpatti, Simon Cavegn,

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Michael wrote:

I found:

* 3# + 5 is prime (11),

* 5# + 7 is prime (37),
* 17# + 19 is prime (510529),
* 59# + 61 is prime (1922760350154212639131).
Using OEIS A035346, the next term would involve the primorial of prime p > 496849.

 

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Paul wrote:

I wasn't able to find another twin prime example upto prime 81799, however there were a few more if we allow P and P+2, or simple prime pairs. Here are my findings:

Prime Pair: 2, 3 -> 2 + 3 = 5 (Prime)
Prime Pair: 3, 5 (Twin Pair) -> 6 + 5 = 11 (Prime)
Prime Pair: 5, 7 (Twin Pair) -> 30 + 7 = 37 (Prime)
Prime Pair: 13, 17 -> 30030 + 17 = 30047 (Prime)
Prime Pair: 17, 19 (Twin Pair) -> 510510 + 19 = 510529 (Prime)
Prime Pair: 19, 23 -> 9699690 + 23 = 9699713 (Prime)
Prime Pair: 43, 47 -> 13082761331670030 + 47 = 13082761331670077 (Prime)
Prime Pair: 53, 59 -> 32589158477190044730 + 59 = 32589158477190044789 (Prime)
Prime Pair: 59, 61 (Twin Pair) -> 1922760350154212639070 + 61 = 1922760350154212639131 (Prime)
Prime Pair: 73, 79 -> 40729680599249024150621323470 + 79 = 40729680599249024150621323549 (Prime)
Prime Pair: 367, 373 -> Digit Length: 149 (Prime)
Prime Pair: 6143, 6151 -> Digit Length: 2628 (Prime)
Prime Pair: 17099, 17107 -> Digit Length: 7360 (Prime)
Prime Pair: 30893, 30911 -> Digit Length: 13311 (Prime)
Prime Pair: 32341, 32353 -> Digit Length: 13927 (Prime)
Prime Pair: 32713, 32717 -> Digit Length: 14108 (Prime)
Prime Pair: 41233, 41243 -> Digit Length: 17802 (Prime)
Prime Pair: 81649, 81667 -> Digit Length: 35332 (Prime)

 

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Oscar wrote:

I've checked primality of p#+q for pairs of consecutive primes (p,q) among the first 5000 primes. Sixteen pairs generate a prime (or at least a PRP), four such pairs involve twin primes.

p  q

2  3

3  5  (twin)

5  7  (twin)

13  17

17  19  (twin)

19  23

43  47

53  59

59  61  (twin)

73  79

367  373

6143  6151

17099  17107

30893  30911

32341  32353

32713  32717

 

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Simon wrote:

3# + 5 is prime 11
5# + 7 is prime 37
17# + 19 is prime 510529
59# + 61 is prime 1922760350154212639131

No more results found.
Searched up to the 29667th prime. The number to check has 150092 digits.

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