Chapter Questions
Mark the following statements true $(T)$ or false $(F)$ justifying your answer briefly.a) Let $N(>1)$ be a given positive integer. All the positive divisors of $N$ can be determined from the prime factorization of $N$.b) Let $p$ be an odd prime and $n>1$. Then, $p^n+1$ is not a square.c) Let $t$ be an even integer. Suppose that a,b,c are integers having no common factor $>1$, then, it is impossible to choose $a, b, c$ such that
$$t^2=a^2+b^2+c^2$$
d) If $n \equiv 3$ or $6(\bmod 9)$, then $n$ is not representable as a sum of two squares.e) Let $r$ be a square-free integer. It is certain that an abelian group of order $r$ is cyclic.f) The matrix ring $M_2(\mathbb{Z} / 2 \mathbb{Z})$ has proper two-sided ideals. That is, $M_2(\mathbb{Z} / 2 \mathbb{Z})$ is not a simple ring.
For any $a \in \mathbb{Z}$, show that $a^5-a \equiv 0(\bmod 5)$.
Find the integral solutions of $6 x+4 y=14$.
Let $a_1, a_2, \cdots, a_t$ be integers. We define
$$I=\left\{a_1 x_1+a_2 x_2+\cdots+a_t x_t: x_i \in \mathbb{Z}, \quad i=1,2, \cdots, t\right\} .$$
Show that I is an ideal of $\mathbb{Z}$, generated by g.c.d $\left(a_1, a_2, \cdots, a_t\right)$.
Find all integers a such that for $0<a<13$,
$$x^2=a(\bmod 13)$$
has a solution.
Let $\left\{c_1, c_2, \cdots, c_t\right\}$ where $t=\phi(r)$ be a reduced residue system $\bmod r$. Show that
$$c_1+c_2+\cdots+c_t \equiv 0 \quad(\bmod r) .$$
Solve the congruence: $45 x \equiv 36(\bmod 54)$.
Solve the Diophantine equation: $6 x+15 y=6$.
Solve the Diophantine equation: $8 x+3 y=28$.
Let p be an odd prime. Suppose that g.c.d $(a, p)=1$. Show that the congruence
$$x^2 \equiv a\left(\bmod p^t\right)$$
has either no solutions or exactly two solutions modulo $p^t$.
A Pythagorean triple is a triple $(a, b, c)$ of positive integers such that $a^2+b^2=c^2$. Let $c>0$. Show that there is a Pythagorean triple $(a, b, c)$ if, and only if, $c$ is divisible by some prime $p$ with $p \equiv 1(\bmod 4)$.
We write $A_{700}(7)=\{[x] \in \mathbb{Z} / 700 \mathbb{Z}:[7][x]=[0]\}$.
Show that $A_{700}(7)$ is a field isomorphic to $\mathbb{Z} / 7 \mathbb{Z}$.
Find the least positive residue of (i) $2^{32}$ (ii) $2^{47}$ modulo 47.
Let $f_n=2^{2^n}+1 . f_n$ is called a Fermat number. If $f_n$ is a prime, it is called a Fermat prime. For $m \neq n$, show that $f_m$ and $f_n$ are relatively prime to one another.(Gauss showed that a regular p-gon can be constructed with a ruler and compass for those values of prime $p$ for which $p=f_n ; n=2$ gives $p=17$.)
Solve the congruence $71 x \equiv 4(\bmod 55)$.
Let $n$ be a product of four consecutive positive integers. Prove or disprove: $(n+1)$ is a perfect square.
(Landau) Let $m \equiv 5(\bmod 12)$ and $m>17$. Show that $m$ is expressible as a sum of three distinct positive squares.
Let $r=x^2+y^2$ where g.c.d $(x, y)=1$. If $d(\geq 1)$ is a divisor of $r$, show that $d$ is also a sum of two squares.
(Ethan D. Bolker). Let $r=p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ where $p_1, p_2, \ldots, p_k$ are distinct primes. We denote the (symmetric) group of permutations on $n$ symbols by $S_n$. Show that the least value of $n$ for which $S_n$ contains an element of order $r$ is $n=p_1^{a_1}+p_2^{a_2}+\cdots+p_k^{a_k}$.
(Thue) Suppose that $r$ is not a perfect square. Let $t \in \mathbb{Z}$. Show that the congruence $t x \equiv y(\bmod r)$ has a solution $<x, y>$ in which $|x|$ and $|y|$ are both $<\sqrt{r}$ and $<x, y>\neq<0,0>$.