Problems & Puzzles: Puzzles

Puzzle 1162  n*2^(2n + 1) + 1 Sebastian Martin Ruiz. sent the following nice puzzle. He comprobado que la siguiente expresión es compuesta desde n=1 hasta 6000   n*2^(2n + 1) + 1 Sin factores triviales, aunque muchos de ellos son divisibles por 2n+1 o un divisor de 2n+1 Q) Hallar un primo de esta forma o probar que no existe

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From 10 to 16 Feb 2024, contributions came from Michael Branicky, Emmanuel Vantieghem, Giorgos Kalogeropoulos, V. F. Izquierdo, Alessandro Casini, Gennady Gusev

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Michael Branicky wrote:

It is prime for n = 9248.

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Emmanuel Vantieghem wrote:

About Puzzle 1162  I found that the expression

   n*2^(2n+1) + 1

is prime when  n = 9248.

I used Mathematica and Dario Alpern's Alpertron.

 

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 Giorgos Kalogeropoulos wrote:

for n=9248 we get a 5573-digits prime.

for n=16146 we get a 9726-digit prime

I will stop my search here

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V. F. Izquierdo wrote:

We have  r = n*2^(2n + 1) + 1

 

If n is of the form:

      10k+2  then r is multiple of 5

      10k+3  then r is multiple of 5

      3k+1   then r is multiple of 3

 

r is Prime with n=9248 o n=16146 for all n <= 25000.

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Alessandro Casini wrote:

This sequence is essentially that of C_(2n), i.e. Cullen numbers with even indexes. Hence, any even value in OEIS A005849 gives a prime of that form. For example, n = 9248, 16146, 29828, ...

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Gennady Gusev wrote:

There were found 3 primes for n<=60000:

 

9248*2^(2*9248+1)+1 is prime! [N-1, Brillhart-Lehmer-Selfridge] (digits:5573)

16146*2^(2*16146+1)+1 is prime! [N-1, Brillhart-Lehmer-Selfridge] (digits:9726)

29828*2^(2*29828+1)+1 is prime! [N-1, Brillhart-Lehmer-Selfridge] (digits:17964)

 

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