Problems & Puzzles:
Reserved for the Gods?
JM Bergot sent this nice puzzle:
Find couples of prime numbes (p, q) such that:
p|Sq & q|Sp
Bergot reports only one of these
couples: (151, 809).
Later he added "have been informed that the
first solution was found in one second, but NO other solution was
found after several hours of execution time... .This puzzle may have
just one solution OR to find the 2nd solution is reserved for the
Can you get the next three couples
reserved primes (p, q)?
Contributions came from J. K. Andersen, Wilfred Whieside
A007506 is Primes p with property that p divides the sum of all
primes <= p.
If p=q was allowed then A007506 would be solutions: 2, 5, 71,
I saw puz486 and gave it a try. But no luck. Below are my
All distinct prime pairs p,q were tested up to 200,000,000
Side note: n=199,999,991 was the max prime in this range with Sn =
Only the already posted solution was found:
p=151, q=809 with Sp=2427, Sq=50887 found where (Sq%p) = (Sp%q) = 0
The following near misses where found where (Sq%p)<=1 and (Sp%q)<=1
p=3, q=5, Sp=5, Sq=10 yields (Sq%p)=1 and (Sp%q)=0
p=349, q=5443, Sp=10887, Sq=1814801 yields (Sq%p)=1 and (Sp%q)=1
p=19853, q=1042609, Sp=20852181, Sq=40717887557 yields (Sq%p)=0 and
p=27827, q=496901, Sp=39752081, Sq=9794603115 yields (Sq%p)=1 and (Sp%q)=1
p=38803, q=421913, Sp=74678602, Sq=7147046965 yields (Sq%p)=1 and (Sp%q)=1
p=73043, q=116507, Sp=249557995, Sq=607352546 yields (Sq%p)=1 and (Sp%q)=1
I would think that there should be an infinite number of solutions
that the odds of any given p,q both being prime and satisfying Sq%p
= Sp%q = 0
would go like 1/(q*p*ln(q)*ln(p)) which has an infinite double
integral as q,p
go to infinity. It just climbs very slowly and the solutions would
separated by exponentially increasing orders of magnitude. Anyhow, I
anyone finds another solution, but I'd be happy to be wrong.