Use Euclid's proof to show that there are infinitely many primes of the form 4n — 1.
Prove that there are infinitely many primes of the form 4n — 1. (Hint: Modify Euclids proof that there are infinitely many primes. First, prove that any number of the form 4n — 1 has a prime factor which is of the form 4k — 1.)
Sum of Newton binomial coefficients with intermittent sign
Sum of Newton binomial coefficients with intermittent sign. See attached file for full problem description.
Show that (2 + √2)^n + (2 - √2)^n is an integer for all positive integers n by using the binomial theorem.
Polynomials, Integers and Divisibility
Problem 6. Suppose that f(s) = adxd+ad-1xd... is a polynomial with integral coefficients (so a0, a1 . . ,ad E Z). Show that f(n)—f(m) is divisible by n — m for all distinct integers n and m.
Number Theory. 400 level. Introductory Course in Undergraduate.
Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. (See attached file for full problem description)
The following geometric arrays suggest a sequence of numbers: 2 6 12 20 a)Find the next three terms. b)Find the 100 term. c)Find the nth term.
Congruence and Euler's Theorem : Use Euler's Theorem to find the last two digits of 7^1245.
Use Euler's Theorem to find the last two digits of 7^1245. Please see the attached file for the fully formatted problem.
Congruence Classes : Prove that a square is always congruent to 0, 1, or 4 modulo 8.
Prove that a square is always congruent to 0, 1, or 4 modulo 8. Please see the attached file for the fully formatted problem.
Show that 2006 is not the difference between two squares.
Show that 2006 is not the difference between two squares. Please see the attached file for the fully formatted problem.
Prove that an integer n is divisible by 125 if and only if the last three digits of n are 000, 125, 250, 375, 500, 625, 750 or 875.
