Question 87
In this problem, we will study how many factors 2 appear in the factorization of numbers of the form $5^{2^{n}}-1$.
a. Let $x$ and $y$ be two odd integers. Prove that $x^{2}+y^{2}$ has exactly one factor 2 in its prime factorization.
b. Using the factorization $a^{2}-b^{2}=(a-b)(a+b)$, determine how many factors 2 the number $5^{4}-1$ has.
c. How many factors 2 does the number $N=5^{2^{2017}}-1$ have?
d. Given that the number $5^{20}$ has 14 digits, prove that $5^{2^{18}+20}$ has 6 consecutive zeros in its decimal representation.
a. Let $x$ and $y$ be two odd integers. Prove that $x^{2}+y^{2}$ has exactly one factor 2 in its prime factorization.
b. Using the factorization $a^{2}-b^{2}=(a-b)(a+b)$, determine how many factors 2 the number $5^{4}-1$ has.
c. How many factors 2 does the number $N=5^{2^{2017}}-1$ have?
d. Given that the number $5^{20}$ has 14 digits, prove that $5^{2^{18}+20}$ has 6 consecutive zeros in its decimal representation.