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 Checking the quantity (Posted on 2023-09-23)

p^a+q^b=r^c

How many distinct solutions of the equation above are there, subject to the following constraints:

p, q, & r distinct primes
a, b, & c distinct positive integers,
each more than one
None of the powers exceeds 1111.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 my computer findings | Comment 2 of 8 |
pr=sym(primes(100));
for w1=1:length(pr)
p=pr(w1);
for w2=w1+1:length(pr)
q=pr(w2);
for a=2:11
for b=2:11
if a~=b
rhs=p^a+q^b;
f1=factor(rhs);
f=unique(f1);
if length(f)==1 && length(f1)>1
disp([p a q b f length(f1)])
end
end
end
end
end
end

finds

[2, 4, 3, 2, 5, 2]
[2, 5, 7, 2, 3, 4]
[2, 7, 17, 3, 71, 2]
[7, 3, 13, 2, 2, 9]

before being cut off manually

meaning

2^4 +3^2=5^2     b and c not distinct here -- ignore
2^5 +7^2=3^4
2^7+17^3=71^2
7^3+13^2=2^9

 Posted by Charlie on 2023-09-23 20:05:21

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