Always composite?

Primes, divisors, arithmetic, number properties, ...
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Alhazen
Posts: 94
Joined: Sun Feb 20, 2011 9:55 pm

Always composite?

Post by Alhazen »

a is a positive integer>1

Let k=a^(a+1)+(a+1)^a

Examples :

k=2^3+3^2=8+9=17 is prime
k=3^4+4^3=145 is composite

If a>2 k is it always composite ?

If yes can we prove it?

Thank you for any comment.
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Lord_Farin
Posts: 239
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Location: Netherlands

Re: Always composite?

Post by Lord_Farin »

k = 80, 342 and 848 yield primes (at least, that's what Mathematica says).
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Alhazen
Posts: 94
Joined: Sun Feb 20, 2011 9:55 pm

Re: Always composite?

Post by Alhazen »

Lord_Farin wrote:k = 80, 342 and 848 yield primes (at least, that's what Mathematica says).
Thank you very much.
You helped me to find the list in Sloane

https://oeis.org/search?q=2%2C80%2C342% ... &go=Search

A073499 as a simple table:
n a(n)
1 1
2 2
3 80
4 342
5 848
6 1194
7 2658
8 4790
9 9376

What is strange is that all are even.
Is that means that if k is odd then there will be no prime of such form?
Thank you.
There is maybe some secret to discover here :D
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