Can you do #8? ### Educational Resource: Advanced Number Theory Exercises#### Use Fermat’s Theorem to compute the following quantities:1. \( 31^{100} \mod 19 \)2. \( 2^{10000} \mod 29 \)3. \( 99^{99} \mod 31 \)#### Prove or demonstrate the following statements:1. Show that \( 11^{84} - 5^{84} \) is divisible by 7.2. Show that if \( n \equiv 2 \pmod{4} \), then \( 9^n + 8^n \) is divisible by 5.3. For which values of \( n \) is \( 3n^2 + 2^n \) divisible by 13? By 7?4. Use Fermat’s Theorem to show that \( n^{13} - n \) is divisible by 2730 for all \( n \).5. Show that if \( p > 3 \) is prime, then \( ab^p - ba^p \) is divisible by \( 6p \).6. Show, using the Binomial Theorem, that if \( p \) is prime and \( a \) and \( b \) are integers, then \( (a+b)^p \equiv a + b \pmod{p} \).7. Show that no prime number of the form \( 4k+3 \) can divide a number of the form \( n^2 + 1 \).8. Show that there are infinitely many primes of the form \( 16k + 1 \). More generally, show that for any \( r > 0 \), there are infinitely many primes of the form \( 2^r k + 1 \).9. Let \( n = r^4 + 1 \). Show that 3, 5, and 7 cannot divide \( n \). What is the smallest prime that can divide \( n \)? Determine the form of the prime divisors of \( n \).10. Show that any proper factor, whether prime or not, of a composite Mersenne number \( 2^p - 1 \) is of the form \( 1 + 2pk \) for some \( k \).11. What can you say about the prime factors of a composite Fermat number \(

Name: Can you do #8? ### Educational Resource: Advanced Number Theory Exerci
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Can you do #8? ### Educational Resource: Advanced Number Theory Exercises#### Use Fermat’s Theorem to compute the following quantities:1. \( 31^{100} \mod 19 \)2. \( 2^{10000} \mod 29 \)3. \( 99^{99} \mod 31 \)#### Prove or demonstrate the following