Let us divide the odd positive integers into two arithmetic progressions; the red numbers are 1, 5, 9, 13, 17, 21, ... The blue numbers are 3, 7, 11, 15, 19, 23,.... (a) Prove that the product of two red numbers is red, and that the product of two blue numbers is red. (b) Prove that every blue number has a blue prime factor. (c) Prove that there are infinitely many blue prime numbers. Hint: Follow Euclid’s proof, but multiply a list together, multiply the result by four, then subtract one.

Let us divide the odd positive integers into two arithmetic progressions; the red numbers are 1, 5, 9, 13, 17, 21, ... The blue numbers are 3, 7, 11, 15, 19, 23,.... (a) Prove that the product of two red numbers is red, and that the product of two blue numbers is red. (b) Prove that every blue number has a blue prime factor. (c) Prove that there are infinitely many blue prime numbers. Hint: Follow Euclid’s proof, but multiply a list together, multiply the result by four, then subtract one.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 92E

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Question

Let us divide the odd positive integers into two arithmetic progressions; the red numbers are 1, 5, 9, 13, 17, 21, ... The blue numbers are 3, 7, 11, 15, 19, 23,....

(a) Prove that the product of two red numbers is red, and that the product of two blue numbers is red.

(b) Prove that every blue number has a blue prime factor.

(c) Prove that there are infinitely many blue prime numbers. Hint: Follow Euclid’s proof, but multiply a list together, multiply the result by four, then subtract one.

Expert Solution