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# Number Theory

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What is the product of the two smallest prime factors of 2^{100} - 1?

May 26, 2021

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What is the product of the two smallest prime factors of $$2^{100} - 1$$?

First prime factor 2 ?
$$2^{100} - 1$$ is an odd number so 2 is no prime factor in $$2^{100} - 1$$

next prime factor 3 ?
If 3 is a prime factor in $$2^{100} - 1$$, then $$2^{100} - 1 \equiv 0 \pmod{3}$$
$$\begin{array}{|rcll|} \hline 2^{100} - 1 &\equiv& 0 \pmod{3} \quad ? \quad | \quad 2 \equiv -1 \pmod{3} \\ (-1)^{100} - 1 &\equiv& 0 \pmod{3} \quad ? \\ 1-1 &\equiv& 0 \pmod{3} \quad ? \\ 0&\equiv& 0 \pmod{3}~ \checkmark \quad | \quad {\color{red}3} ~\text{is a prime factor in}~ 2^{100} - 1 \\ \hline \end{array}$$

next prime factor 5 ?
If 5 is a prime factor oin $$2^{100} - 1$$, then $$2^{100} - 1 \equiv 0 \pmod{5}$$
$$\begin{array}{|rcll|} \hline 2^{100} - 1 &\equiv& 0 \pmod{5} \quad ? \\ 2^{2*50} - 1 &\equiv& 0 \pmod{5} \quad ? \\ \left(2^2\right)^{50} - 1 &\equiv& 0 \pmod{5} \quad ? \quad | \quad 2^2=4 \equiv -1 \pmod{5} \\ (-1)^{50} - 1 &\equiv& 0 \pmod{5} \quad ? \\ 1-1 &\equiv& 0 \pmod{5} \quad ? \\ 0&\equiv& 0 \pmod{5} ~ \checkmark \quad | \quad {\color{red}5} ~\text{is a prime factor in}~ 2^{100} - 1 \\ \hline \end{array}$$

$$3*5 = 15$$

The product of the two smallest prime factors of $$2^{100} - 1$$ is 15

May 26, 2021
edited by heureka  May 26, 2021
edited by heureka  May 26, 2021