Determine all possible triplets (p, q, n) of nonnegative integers that satisfy this equation:
2p + 2q = n!
Provide valid argument for your answer.
*** Adapted from a problem which appeared at the Harvard/MIT math Olympiad.
f=1;
for i=2:17
f(end+1)=f(end)*i;
end
for p=0:48
for q=0:48
nf=2^p+2^q;
if ismember(nf,f)
n=find(f==nf);
disp([p q n])
end
end
end
finds 5 triplets:
0 0 2
1 2 3
2 1 3
3 4 4
4 3 4
This tested every n through 17 for being able to be the total. Possibly there could be a proof that factorials above 17! (or even lower) cannot be the sum of two powers of 2.
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Posted by Charlie
on 2023-02-27 12:46:52 |
without proof of completeness
