MathLabs

Problem 4

Determine all positive integers relatively prime to all the terms of the infinite sequence an=2n+3n+6n−1, n≥1.a_n=2^n+3^n+6^n -1,\ n\geq 1.
Step 2 of 5: Fermat's little theorem for primes p>3p>3
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p>3 prime ⟹ 2p−1≡3p−1≡6p−1≡1(modp)p>3\ \text{prime}\ \Longrightarrow\ 2^{p-1}\equiv3^{p-1}\equiv6^{p-1}\equiv1\pmod p
Detailed analysis

For any prime p>3p>3, none of 2,3,62,3,6 is divisible by pp, so Fermat's little theorem gives 2p−1≡12^{p-1}\equiv1, 3p−1≡13^{p-1}\equiv1, 6p−1≡1(modp)6^{p-1}\equiv1\pmod p.