Problems & Puzzles: Puzzles

 Puzzle 442. Primes ending in 9 JM Bergot asked recently for primes ending in 9 such that the following nice pattern gives other primes: 89*2 + 1=179 89*3 + 2=269 89*4 + 3=359 89*5 + 4=449 89*6 + 5=539 Fail, not prime. And asked by larger patterns: I got a such a larger pattern of primes with the starting prime 407874179: 407874179*2+1 = 815748359 407874179*3+2 = 1223622539 ... 407874179*9+8 = 3670867619 407874179*10+9 Not prime Q1. Get larger patterns.

Contributions came from J. K. Andersen, Andrea Concaro & Seiji Tomita.

J. K. Andersen wrote:

If the first prime is p then the pattern with n primes is p*k+(k-1) = (p+1)*k-1, for k = 1, 2, 3, ..., n. If p>5 and n>4 then p must end in the digit 9.

The smallest value of p+1 for n up to 13 is in http://www.research.att.com/~njas/sequences/A088651. The value of p for n = 10, 11, 12, 13: 214580145779, 9448481062019, 247236503934419, 2545206711847799.

The same form (except k=1 did not have to give a prime) was searched in a follow-up to puzzle 379.

The smallest prime for n = 14 is 18178612369988250179.

The smallest for n = 15 is 53792264108455702829

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Andrea wrote:

Larger patterns start at primes:

214580145779 (lowest starting prime of a 10 elements pattern)
9448481062019 (lowest starting prime of a 11 elements pattern)

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Seiji Tomita wrote:

I looked for p that p*n+n-1 was a prime number under the p < 3*10^10. I found such primes p that p*n+n-1 were prime in the range 2<=n<=9.
p=1674689729,6380217479,28081637219,15002412599,and 24291715139

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