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

***

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.

***

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

***

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 .

***

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.

***

See more solutions on this topic at Puzzle 851

***