Is 98765
4 + 4
56789 a prime number?
Provide your answer with valid reasoning.
98765^4 mod 17 = 13
4^n goes in a length-4 cycle, mod 17, of 4, 16, 13, 1, ...
56789 is congruent to 1 mod 4, so 4 to that power is congruent to 4 mod 17
98765^4 + 4^56789 is therefore contruent to 13 + 4 mod 17, or 0 mod 17, and is thus divisible by 17, and is not prime.
A more direct approach (and what led me to the above), via computer:
a=sym(98765)^4;
b=sym(4)^56789;
n=a+b;
p=1;
for i=1:10000
p=nextprime(p);
if mod(n,p)==0
disp(p)
n=n/p;
else
p=p+1;
end
end
finds that of the first 10,000 prime numbers, the given sum is divisible by:
17
41
61
521
733
769
42169
This is a long way from completely factoring that 34191-digit number but it does show it's composite.
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Posted by Charlie
on 2023-05-23 11:56:58 |
solution
