Problems & Puzzles: Puzzles

Puzzle 379. SG primes and its prime average. Jaime Ayala asks for primes triplets (p, r, q) such that: q=2p+1 & r=(p+q)/2. Evidently r is the average of the SG couple p & q, and in general r=(3p+1)/2. The smallest triple of this type is (3, 5 & 7). A moderate large (100 digits) example is gotten with the prime p=10^99-1 +  4*1439363. J. K. Andersen notices that these triplets can be expressed in this nice way: (2n-1, 3n-1, 4n-1). If you are looking for large provable primes perhaps is better you use certain prime numbers models. For example: p=k*2^n-1; q=k*2^(n+1)-1; r=3*k*2^(n-1)-1 Questions:  A)   Find a Titanic triplet (p = 1000 digits) B)     Find a  Gigantic triplet (p=10000 digits)

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Contributions came from Farideh Firoozbakht & C. Rivera.

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Farideh wrote

The largest triplet (219 digits) of the form (k* 2^n-1, 3k* 2^(n-1)-1, k*2^(n+1)-1) that I found is for n=700 & k=36193095...

 

Another large triplet (249 digits) of the form (k* 2^n-1, 3k* 2^(n-1)-1, k*2^(n+1)-1)

is for n=802  & k=11561283.

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BTW, exploring and extending the description of these triplets given above by J. K. Andersen, I asked myself for k-tuplets of primes such that some k consecutive members of the sequence {2n-1, 3n-1, 4n-1, ...} are primes for a given (even) n.

Soon and easily I found a 6-tuplet with n=120: {2n-1, 3n-1, ..., 7n-1} are primes.

Other 6-tuplet is obtained if n=30: {12n-1, 13n-1, ..., 17n-1} are primes.

Perhaps you would like to discover larger prime k-tuplets of this type?

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Luke Pebody wrote (Dec. 8, 2006):

The smallest n such that 2n-1, 3n-1, 4n-1 are prime is 2.
The smallest n such that 2n-1, 3n-1, 4n-1 and 5n-1 are prime is 90.
Throw in 6n-1 and you get n=120, which also has 7n-1 prime.
Throw in 8n-1 and you have to go up to n=2894220.
Throw in 9n-1 and the smallest n is at least 10^8.

Other ranges:
[3..4] - 2
[3..5] - 6
[3..6] - 18
[3..7] - 120
[3..8] - 1260
[3..9] - 1485540
[3..10] - 28667100
[3..11] - 28667100
 

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Vladimir Trushkov wrote (Dec., 06)

The smallest n such that 2n-1, 3n-1, 4n-1, ..., 9n-1 are prime is
397073040.

The smallest n such that 2n-1, 3n-1, 4n-1, ..., 10n-1 are prime is
1236161850.

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Farideh wrote (28 Dec. 2006):

 

The smallest n such that all ten numbers 2n-1, 3n-1, 4n-1, 5n-1, 6n-1, 7n-1, 8n-1

9n-1, 10n-1 & 11n-1 are prime is 764907546690.

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Farideh Firoozbakht has extended the Pebody results:

[3..12] - 842889105240
[3..13] - 2281585556250

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JK Andersen wrote (March, 07)

The smallest titanic triplet has p = 10^999 + 20364505399.
PrimeForm/GW found the prp's. Marcel Martin's Primo proved p and r.
PrimeForm/GW quickly proved q = 2p+1 by using p.

A gigantic triplet looks around (10000/1000)^5 = 100000 times harder.
Far too hard for me.
 

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