Number Theory
Elementary Number Theory Chapter 3.3 Page 54, Why choose "$N = 4q_{1}q_{2} \cdots q_{s} - 1$"?(David M. Burton's 7th Edition) Paid Members Public
Why does the author choose $N = 4q_{1}q_{2} \cdots q_{s} - 1$ when proving there are an infinite number of primes of the form $4n + 3$.
Elementary Number Theory Chapter 3.3 Page 54, Why "But $r_{i}$ cannot be found among the listing $q_{1}, q_{2}, ...$"?(David M. Burton's 7th Edition) Paid Members Public
Elementary Number Theory Chapter 3.3 Page 54, Why "But $r_{i}$ cannot be found among the listing $q_{1}, q_{2}, ..., q_{s}$, for this would lead to the contradiction that $r_{i} \mid 1$"?
Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q12 Paid Members Public
My Solution for "Let $p_{n}$ denote the $n$th prime number. For $n \geq 3$, prove that $p_{n+3}^{2} < p_{n}p_{n+1}p_{n+2}$. [Hint: Note that $p_{n+3}^{2} < 4p_{n+2}^2 <8p_{n+1}p_{n+2}$.]"
Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q11 Paid Members Public
My Solution for "Find the smallest positive integer $n$ for which the function $f(n) = n^2 + n + 17$ is composite. Do the same for the functions $g(n) = n^2 + 21n + 1$ and $h(n) = 3n^2 + 3n + 23$."
Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q10 Paid Members Public
My Solution for "Establish that the sequence $$(n + 1)! - 2, (n + 1)! - 3, ... , (n + 1)! - (n + 1)$$ produces $n$ consecutive composite integers for $n > 2$."
Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q9 Paid Members Public
My Solution for "(a) For $n > 3$, show that the integers $n$, $n + 2$, $n + 4$ cannot all be prime. (b) Three integers $p$, $p + 2$, $p + 6$, which are all prime, are called a $\textit{prime-triplet}$. Find five sets of prime-triplets."
Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q8 Paid Members Public
My Solution for "Given a positive integer $n$, it can be shown that there exists an even integer $a$ that is representable as the sum of two odd primes in $n$ different ways. Confirm that the integers $60$, $78$, and $84$ can be written as the sum of two primes in 6, 7, and 8 ways, respectively."
Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q7 Paid Members Public
My Solution for "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 $75$."
