MATH 400c Topics In Math-number Theory Colorado State

Mathematics 400c Homework (due Apr. 7) A. Hulpke 56) Let p be prime with p 3 (mod 4) and let q = 2p + 1. q1 a) Show that if q is prime, we have that q|2 p 1. (Hint: Evaluate 2 2 mod q.) b) Show (without using a computer) that 250051 1 is compos

Math 400c, MWF 12:00, E 206 Lecturer: Alexander Hulpke, Weber 217 Ofce Hours: MTF2, W3 preliminary. See http:/www.math.colostate.edu/hulpke/ officetimes.html Email: hulpke@math.colostate.edu WWW: http:/www.math.colostate.edu/hulpke/lectures/m400c Pho

Mathematics 400c Homework (due Mar. 10) A. Hulpke 37) 38) (GAP) Find a primitive root modulo 73. Solve the equation 6x 32 (mod 73). Determine (2268). Find a number a such that ord2268 (a) = (2268). 39) Find all solutions to the following equati

Mathematics 400c Homework (due April 28) A. Hulpke 71) This problem is to give an alternative proof of the innitude of primes due to EULER: Assume there would be only nitely many primes. Show that the product formula for the value (1) equals a div

Mathematics 400c Homework (due April 21) A. Hulpke 66) Let f be a function of a real variable. Dene F(x) = nx f (x/n) (n N) Show (general Mobius inversion formula) that f (x) = nx (n)F(x/n) 67) Show that (x) = (n) (n 1) log n 2nx

Mathematics 400c Homework (due Feb. 4) A. Hulpke 7) Find a sequence of 99 consecutive integers which are not prime. 8) Compute (without using a Gcd function on a computer) the Gcd of a = 67458 and b = 43521 and express it in the for xa + yb. 9) Th

Mathematics 400c 1) Homework (due Jan. 28) A. Hulpke Show, that if there are integes r and s, such that a = r 2 s2 , b = 2rs, c = r2 + s2 then (a, b, c) is a pythagorean triple, i.e. a2 + b2 = c2 . 2) Prove, that the product of two consecutive i

Mathematics 400c Homework (due Feb. 25) A. Hulpke 25) Show that 1729 = 7 13 19 is a Carmichael number. Side remark: One of the best-known anecdotes in the history of mathematics is about a visit that G.H.Hardy (for more historical information s

Mathematics 400c Homework (due Feb. 18) A. Hulpke 19) Let p be a prime. Show that a p + b p (a + b) p (mod p). Hint: Consider the expansion of (a + b) p according to the binomial theorem. 20) a) Determine a number x such that x2 15 (mod 113 ). b