Problems & Puzzles: Puzzles

 Puzzle 1125 About Feb 28   February 28 is the 59th day of the year in the Gregorian calendar. Exactly last Tuesday a dear friend of mine (Ernesto) sent to me the following nice curio: 5^59-4^59, 4^59-3^59 and 3^59-2^59 are primes. Q1. Can you find a better sequence for 59 than this? Q2. What about another integer but 59?

During the week from March 4-10, 2023, contributions came from Michael Branicky, Alain Rochelli, Jeff Heleen, Giorgos Kalogeropoulos, Oscar Volpatti

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Michael wrote:

Denoting n as the exponent and k the starting integer raised to that power, I found one sequence of length 7, with n = 7, k = 77664240: 77664241^7 - 77664240^7 is prime 77664242^7 - 77664241^7 is prime 77664243^7 - 77664242^7 is prime 77664244^7 - 77664243^7 is prime 77664245^7 - 77664244^7 is prime 77664246^7 - 77664245^7 is prime 77664247^7 - 77664246^7 is prime Below are selected others, reported as (sequence length, n, k) and, with only the smallest k for a given sequence length and n reported: (6, 5, 10291135) (5, 5, 1915328) (5, 7, 14020) (5, 19, 5421699) (4, 3, 1) (4, 5, 382) (4, 7, 6) (4, 29, 6) (4, 11, 510811) (4, 13, 81605) (4, 19, 483904) I found no longer sequence for n = 59, trying k = 1..10^7. However, the following values of k produce others of length 3: k = 416746, 567072, 579311, 1194762, 2161020,
2472300, 4099705, 4233560, 4264670, 4441903,
4789298, 6142986, 6901797, 7477964, 8162358, 8393122

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Alain wrote:

I did not get better sequence of 3 primes with an exponent > 59.

I got the following sequences of 4 primes:

5^3-4^3, 4^3-3^3, 3^3-2^3 and 2^3-1^3 are primes

386^5-385^5, 385^5-384^5, 384^5-383^5 and 383^5-382^5 are primes

10^7-9^7, 9^7-8^7, 8^7-7^7 and 7^7-6^7 are primes

222^7-221^7, 221^7-220^7, 220^7-219^7 and 219^7-218^7 are primes

10^29-9^29, 9^29-8^29, 8^29-7^29 and 7^29-6^29 are primes

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Jeff wrote:

The smallest exponent for which I found an example of 4 primes in a row is 7.

7^7 - 6^7 = 543607 = prime
8^7 - 7^7 = 1273609 = prime
9^7 - 8^7 = 2685817 = prime
10^7 - 9^7 = 5217031 = prime

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Giorgos wrote:

Q2. k^7 - (k-1)^7 produces a chain of 7 primes for k = 77664241, 77664242, 77664243, 77664244, 77664245, 77664246, 77664247

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Oscar wrote:

For compactness, I'll list solutions as triplets (k,p,n):
a sequence of k consecutive primes is obtained for exponent p,
from prime  (n+1)^p-(n)^p  to prime  (n+k)^p-(n+k-1)^p.
Hence, the given example would be listed as triplet (3,59,2).

For exponent p=59, I only found some more solutions with length k=3;
second solution: (3,59,416746), with 328 digits;
100-th solution: (3,59,113485815), with 469 digits.

I found some nice solutions with small exponent p and length k<=8.
(3,2,1)
(4,3,1)
(5,5,1915328)
(6,5,10291135)
(7,7,77664240)
(8,7,16049929484)

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