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Problems & Puzzles:
Puzzles
Problem 91.
Counterexamples to the
stronger Conjecture by Feith-Thompson
On Aug 3, 2025, Giorgos
Kalogeropoulos suggested the following challenge to be
published in my pages.
Let A = (p^q - 1) / (p - 1) and B = (q^p - 1) / (q - 1)
where p and q are distinct primes.
We are searching for p and q, distinct primes, such
that A and B are NOT coprimes (they
share a common factor); which means that gcd(A,B) >
1
The statment that A & B are coprimes for any primes
p & q distinct, is known as the stronger version of
the
Feith-Thompsosn Conjecture, (1962). This
stronger version cojecture has been disapproved.
First counter-example case is p=17, q=3313,
gcd(A,B) = 112643, that was found by N. M. Stephens,
in 1971.
54 years has elapsed since the
Stephens countexample was found. Perhaps is time to
find a second one counterexample, if any other
exist.
Q.
Can you find a second counter-example to the
stronger Feith-Thompson Conjecture?
* If you find
no one counterexample perhaps is useful for others
to report the range of you search.
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From Aug 9-16, 2025 contributions came from
Michael Branicky, Vicente Felipe Izquierdo, Jeff Heleen
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Michael wrote:
I searched all prime pairs p, q with 1 < p < q <= 31627 and found no other
counterexample
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Vicente wrote:
I
can't find any other solution within the following limits:
Prime[2] <= p
<=Prime[950]
(7499)
Prime[p+1] <= q <=Prime[10^4] (104729)***
Jeff wrote:
I have searched up to p(488) = 3491 and
q(9592) = 99991.
I found no solutions other than the one already given (p = 17, q = 3313)
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