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Is $1359^{73}+(137^{18}+253^{16}+154^{22})^{19}$ prime or composite? and WHY? For example for $17^{13}+34^{19}+51^{23}$ i do this: $17(17^{12}+2×34^{18}+3×51^{22})$ Then obviously it is a composite number(can devided by 17). or in some question if we determine that the number is even then its composite. But here the number certainly is odd and i cant see a common multiple. I want to see what techniques we could use for this kind of questions. Thank u

Alireza
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    This is one of those introductory problems you might find right after learning some methods in modular arithmetic. Is that accurate to your situation? If so, try simplifying the expression $\pmod p$ for small primes $p=2,3,5,\ldots$. You should be able to answer the problem soon after starting. (as a sidenote, showing that a number in this type of form is prime is highly nontrivial, so in a textbook problem it's usually safe to start with the assumption that it's composite and try to find a prime divisor, again, by modular arithmetic) – Brian Moehring Oct 28 '22 at 21:51
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    The obvious try is to find some prime factor of $(1359) = 3^2 \times 151$ and try to prove that the sum enclosed in parentheses has the same prime factor. If it turns out that the expression is prime, rather than composite, what would be the educational value in posing the problem? – user2661923 Oct 28 '22 at 22:11
  • This looks like a random expression. It is not immediately obvious that the number has an algebraic or aurifeuillan factor. Perhaps it has a small prime factor and that can be easily shown, but even then is the purpose for doing this by hand for such an arbitary expression debatable. – Peter Oct 29 '22 at 08:11
  • If you just wonder whether this is prime, there are countless tools to test this, one of them is factordb. – Peter Oct 29 '22 at 08:12

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Doing arithmetic modulo 3, we get:

$$1359^{73}+(137^{18}+253^{16}+154^{22})^{19}$$ $$= 0^{73}+(2^{18}+1^{16}+1^{22})^{19}$$ $$= 0 + (2^{18}+1+1)^{19}$$ $$= 0 + (1+1+1)^{19}$$ $$= 0 + 0^{19}$$ $$= 0$$

Therefore, $1359^{73}+(137^{18}+253^{16}+154^{22})^{19}$ is divisible by 3, hence composite.

Dan
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