Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q7
Background
All theorems, corollaries, and definitions listed in the book's order:
I will only use theorems or facts that are proved before this question. So you will not see that I quote theorems or facts from the later chapters.
Question
A conjecture of Lagrange ($1775$) asserts that every odd integer greater than $5$ can be written as a sum $p_{1} + 2p_{2}$, where $p_{1}$, $p_{2}$ are both primes. Confirm this for all odd integers through $75$.
Solution
(I wrote a program for this question.)
$7 = 3 + 2 \times 2$
$9 = 3 + 2 \times 3$
$11 = 5 + 2 \times 3$
$13 = 3 + 2 \times 5$
$15 = 5 + 2 \times 5$
$17 = 3 + 2 \times 7$
$19 = 5 + 2 \times 7$
$21 = 7 + 2 \times 7$
$23 = 13 + 2 \times 5$
$25 = 3 + 2 \times 11$
$27 = 5 + 2 \times 11$
$29 = 3 + 2 \times 13$
$31 = 5 + 2 \times 13$
$33 = 7 + 2 \times 13$
$35 = 13 + 2 \times 11$
$37 = 3 + 2 \times 17$
$39 = 5 + 2 \times 17$
$41 = 3 + 2 \times 19$
$43 = 5 + 2 \times 19$
$45 = 7 + 2 \times 19$
$47 = 13 + 2 \times 17$
$49 = 3 + 2 \times 23$
$51 = 5 + 2 \times 23$
$53 = 7 + 2 \times 23$
$55 = 17 + 2 \times 19$
$57 = 11 + 2 \times 23$
$59 = 13 + 2 \times 23$
$61 = 3 + 2 \times 29$
$63 = 5 + 2 \times 29$
$65 = 3 + 2 \times 31$
$67 = 5 + 2 \times 31$
$69 = 7 + 2 \times 31$
$71 = 13 + 2 \times 29$
$73 = 11 + 2 \times 31$
$75 = 13 + 2 \times 31$
Read More: All My Solutions for This Book