 mersenneforum.org Generalized Mersenne Primes
 Register FAQ Search Today's Posts Mark Forums Read 2012-10-27, 16:04 #1 Unregistered   2×139 Posts Generalized Mersenne Primes I believe that I remember a theorem having to do with Generalized Mersenne primes of the form: (a^n+b^n)/(a+b). The theorem states that if (a^n+b^n)/(a+b) is prime, then n must also be prime. Does anyone know a reference that has a proof of this? Thanks.  2012-10-27, 18:18 #2 Batalov   "Serge" Mar 2008 San Diego, Calif. 242438 Posts For even values of n, this doesn't divide. For them, there's no denominator, but the logic below still applies. For odd values of n: Suppose you have a composite n = k*m, with k>=3. Then, (an+bn)/(a+b) = (akm+bkm)/(a+b) = = (ak+bk)/(a+b)*(a(k-1)m - bm*a(k-2)m + b2m*a(k-3)m + ... + b(k-1)m) where (ak+bk)/(a+b) is > 1. QED, composite. For proper Mersenne numbers a+b = 2+(-1) = 1, which makes for an interesting case - there's no denominator. When b=1, an+bn is a proper Fermat number, and for b>1 and odd (prime) n, (an+bn)/(a+b) has the property that you sought for. Last fiddled with by Batalov on 2012-10-27 at 18:24   2012-10-27, 18:28   #3
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

214816 Posts Quote:
 Originally Posted by Unregistered I believe that I remember a theorem having to do with Generalized Mersenne primes of the form: (a^n+b^n)/(a+b). The theorem states that if (a^n+b^n)/(a+b) is prime, then n must also be prime. Does anyone know a reference that has a proof of this? Thanks.
Code:
(15:18)>for(a=1,100,
for(b=a,100,
for(n=1,100,
if(((a^n+b^n)/(a+b))%1.==0 &&isprime((a^n+b^n)/(a+b)) && !isprime(n),
print(((a^n+b^n)/(a+b))","a","b","n);)
)
)
)
Quote:
 14321,6,26,4 280097,12,69,4 4481,14,18,4 134417,24,57,4
shows that n=4 has counterexamples.   2012-10-27, 19:14 #4 Batalov   "Serge" Mar 2008 San Diego, Calif. 101×103 Posts For even values of n (and not coprime a,b), indeed, you can arrive to some convenient cancellations (a+b will have to be 2*gcd(a,b)n, where n will be a power of two) and get an extended Generalized Fermat in disguise. E.g. (6^4+26^4)/(6+26) is xGF'(3,13,4).   2012-10-27, 19:31 #5 Batalov   "Serge" Mar 2008 San Diego, Calif. 101000101000112 Posts P.S. Correction re: 2*gcd(). Not always 2, of course. 2 will frequent, because there are very many prime xGF'(a,b,4) (=(a^4+b^4)/2) with odd a and odd b. Tons. When a+b is 2x, you can finish the counterexample by multiplying a and b by 2x-1 or similar. Ugh, I need some coffee. Last fiddled with by Batalov on 2012-10-27 at 19:33   2012-10-30, 14:47 #6 Unregistered   2·3·1,319 Posts Thanks so much for your replies. Very helpful. Batalov, if I understand your arguments, if n is odd and composite then (a^n+b^n)/(a+b) must also be composite. Can this be extended to any composite n with an odd factor? If so, then only n that are powers of two could possibly result in a prime. I have found some counterexamples and have observed that a and b are never coprime. Can you think of an argument to support this observation about coprimes? Thanks.  2012-10-31, 14:16   #7
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

214816 Posts Quote:
 Originally Posted by Unregistered Thanks so much for your replies. Very helpful. Batalov, if I understand your arguments, if n is odd and composite then (a^n+b^n)/(a+b) must also be composite. Can this be extended to any composite n with an odd factor? If so, then only n that are powers of two could possibly result in a prime. I have found some counterexamples and have observed that a and b are never coprime. Can you think of an argument to support this observation about coprimes? Thanks.
well assume gcd(a,b)=c then the equation comes to:

(a^n+b^n)/(a+b) = c^n*(d^n+e^n)/c*(d+e) =c^(n-1)*(d^n+e^n)/(d+e) so if (d^n+e^n)/(d+e) is integer so is (a^n+b^n)/(a+b) but with a integer divisor >1 so it's not prime.  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post sweety439 sweety439 140 2022-12-20 07:08 Bob Underwood Math 12 2020-10-11 20:01 carpetpool Information & Answers 9 2018-02-24 21:41 carpetpool Miscellaneous Math 1 2017-03-23 23:42 Cyclamen Persicum Math 1 2004-01-30 15:11

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