Problems & Puzzles: Puzzles

 Puzzle 847. Consecutive primes with the same Collatz length. Abhiram R. Devesh, sent the following nice puzzle: Find sets of n consecutive prime numbers that have the same length of Collatz sequence. Abhiram sent the following examples:   n=2 :: [173, 179] with Collatz length = 31 n=3 :: [409, 419, 421] with Collatz length = 40 n=4 :: [409, 419, 421, 431] with Collatz length = 40   I was able to find for n=22 :: [27512549, 27512557, 27512567, 27512587, 27512609, 27512629, 27512633, 27512651, 27512689, 27512731, 27512743, 27512753, 27512759, 27512767, 27512773, 27512831, 27512833, 27512839, 27512843, 27512861, 27512869, 27512879] with Collatz length = 224 Lets take the number 3, the Collatz sequence is given as [3, 10, 5, 16, 8, 4, 2, 1]. It takes 7 steps to reach 1. Hence the collatz length is 7. Q. Would you like to send larger sets of consecutive primes with the same Collatz length?

Contributions came from Emmanuel Vantieghem and Vicente Felipe Izquierdo

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Emmanuel wrote:

I found 41 consecutive primes with Collatz length 190 :
734483273, 734483297, 734483363, 734483383, 734483423, 734483443, 734483467, 734483483, 734483489, 734483531, 734483537, 734483543, 734483549, 734483557, 734483579, 734483591, 734483597, 734483599, 734483653, 734483663, 734483689, 734483693, 734483723, 734483753, 734483773, 734483807, 734483821, 734483843, 734483851, 734483873, 734483887, 734483933, 734483951, 734483999, 734484011, 734484029, 734484041, 734484043, 734484061, 734484071, 734484169

A longer sequence of consecutive primes (if there is one) should have its first prime > 10^9.

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Vicente wrote:

Hasta el Primo[8.000.000] el máximo de consecutivos con la misma longitud es de 33 primos con longitud de Collatz = 149:

79328611,79328653,79328657,79328663,79328687,79328693,79328699,79328723, 79328729,79328759,79328761,79328773,79328779,79328791,79328807,79328867, 79328917,79328929,79328959,79328989,79329013,79329049,79329053,79329059, 79329067,79329071,79329077,79329079,79329101,79329113,79329127,79329167, 79329179

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Michael Hürter wrote on Set 29, 2016:

I found 269 consecutive primes with Collatz length 334 starting with prime 3416702480489. The last prime is 3416702488091 .

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Dmitry wrote on Set 30, 2016:

I found that every number from 2^626+1 to 2^626+15733190 has the same length=5212 in the Collatz sequence (see https://oeis.org/A277109). In this range there are 36386 consecutive probable primes. For primality testing I used 20 iterations of Miller-Rabin test using GMP's mpz_probab_prime_p() function...(What is the contribution from Guo-Gang Gao to these results from you?)...His paper gave me the idea to look at these patterns 2^n+1 for long runs of equal steps.

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See more solutions on this topic at Puzzle 851

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