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Number theory mathematics
1. State and prove Euler's Theorem (generalization of Fermat's Little Theorem). (4 points) 2. Find the last two decimal digits of 453^561. (4 points) 3. Determine, with explanation, whether or not there exists an integer n ...
Continue reading(1) Find the simple continued fraction expansion of: 290/176 . (2) Find the simple continued fraction expansion of square root of 11 (3) Show that [4, 2] = 2 + square root of 6 (4) ...
Continue reading(1) Find p(8) using the recursive property m(n) = m1(n)+m(nm) for n m > 1. (2) Prove that the number of partitions of n into at most m parts is the number of partitions into ...
Continue reading(1) Evaluate the following Legendre symbols: (a) (11/29) (b) (23/61) (2) Let p > 5 be a prime. Show that at least one of 2, 5, 10 is quadratic residue modulo p. (3) ...
Continue reading(1) Determine the order of the following elements: (a) 5 (mod 31) (b) 3 (mod 76) (2) If p is a prime and g is a primitive root modulo p, is -g also a primitive ...
Continue reading(1) Use Fermat's Little Theorem to compute 31^100 (mod 19). (2) For which values of n is 3^n + 2^n divisible by 13? (3) Show that no prime number of the form 4k + 3 can ...
Continue reading(1) Solve the simultaneous congruence: x = 7 (mod 9) x =0 (mod 10) x = 3 (mod 7) (2) (Bhaskara) There are n eggs in a basket. If eggs are removed from the ...
Continue reading(1) Compute the following quantities: (a) 3^29 (mod 31) (b) 19^19 (mod 7): (2) If possible, nd a pair of integers x and y that satisfy the given condition. If it is not possible, ...
Continue readingLet pn denote the nth prime number. Prove that for every n -> Z+, pn+1 <= p1p2..pn+1 (Hint: use ideas from the proof that there are in fintely many primes.)(2) Let a and n be positive integers such that ...
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