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**Exercise 4.

We consider each part separately.

We are asked to compute $2^{375}$ by various exponentiation methods.

We are asked to construct the first $r+1$ levels of the "power tree" as defined in Figure 14 and in Exercise 4.

**Exercise 4.

We are asked to write a MIX program for Algorithm A (the right-to-left binary method for exponentiation) to compute $x^n \bmod w$, where $w$ is the word size, and then to compare it with a serial mult...

Let u(x)=u_nx^n+\cdots+u_0 be reducible over the integers, and let

Let \(u(x)\in \mathbb{Z}[x]\) be irreducible, with \(n=\deg u\) and coefficient height \(H\).

Let u(x)=u_nx^n+\cdots+u_0 be reducible over the integers, and let

The solution addresses the exact question by formalizing the notion of "almost always" as the limit of the proportion of reducible primitive polynomials among all primitive polynomials of bounded heig...

Let $n$ be fixed and let $p \to \infty$.

Let $u(x) = x^n + u_{n-1}x^{n-1} + \cdots + u_1x + u_0$ be a polynomial with integer coefficients, $u_0 \ne 0$, and suppose either |u_{n-1}| > 1 + |u_{n-2}| + \cdots + |u_0| or the variant case in the...

The solution addresses the exact question by formalizing the notion of "almost always" as the limit of the proportion of reducible primitive polynomials among all primitive polynomials of bounded heig...

Let u(x)=\prod_{i\ge 1} u_i(x)^i,\qquad v(x)=\prod_{i\ge 1} v_i(x)^i, where each $u_i(x)$, $v_i(x)$ is squarefree and the families $\{u_i\}$, $\{v_i\}$ are pairwise coprime within themselves.

Let $p$ be an odd prime and $d \ge 1$.

The statement is **false in general**.

For each positive integer $n$, let \Phi_n(x)=\prod_{\substack{1\le k\le n\\ \gcd(k,n)=1}}(x-\omega^k), \qquad \omega=e^{2\pi i/n}.

Let R=\mathbb{F}_p[x]/(q(x)).

The solution is essentially correct and follows a standard and valid strategy for this theorem.

Let $u(x)$ be a random monic polynomial of degree $n$ over $\mathbb{F}_p$.

Let f_n(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0 be a primitive polynomial of degree $n$ with integer coefficients, and let

The solution correctly addresses the exercise.

The solution correctly addresses the exercise.

Let $u(x) = p_1(x)\cdots p_r(x)$ be squarefree, and let $\deg p_i(x) = d_i$ with $\sum_{i=1}^r d_i = n = \deg(u)$.

The solution correctly addresses the exercise.

Let $u(x)$ be a polynomial with integer coefficients that is squarefree over $\mathbb{Z}$.

The solution correctly addresses the exercise.

Let \[ u(x)=u_n x^n+\cdots+u_0=u_n\prod_{j=1}^n(x-\alpha_j), \qquad \|u\|^2=\sum_{j=0}^n |u_j|^2, \qquad

The solution correctly addresses the exercise.

We are asked to reconstruct a binary string given the counts of its consecutive pairs grouped by how many ones they contain.

Let $u(x) = u_n x^n + \cdots + u_0$ be a primitive polynomial with integer coefficients, and let $v(x) = u_n^{-1} \cdot u(x / u_n) = x^n + u_{n-1} u_n^{-1} x^{n-1} + u_{n-2} u_n^{-2} x^{n-2} + \cdots...

The solution correctly addresses the exercise.

The solution correctly addresses the exercise.

The solution correctly addresses the exercise.

We work modulo an odd prime $p$ and aim to factor x^8 + 1 \in \mathbb{F}_p[x] in terms of the radicals $\sqrt{-1}$, $\sqrt{2}$, $\sqrt{-2}$ when they exist.

Let $u(x)=x^8+1.$ We seek the number $r$ of irreducible factors of $u(x)$ modulo a prime $p$.

We are given a black-box quantum operation that acts on a single qubit.

Assume that $u(x)$ is squarefree and satisfies u(x)=p_1(x)p_2(x)\cdots p_r(x), as in (6).

Assume that $u(x)$ is squarefree and satisfies u(x)=p_1(x)p_2(x)\cdots p_r(x), as in (6).

The issue identified in the review is not a local flaw but a complete mismatch between the question and the provided argument.

Assume that $u(x)$ is squarefree and satisfies u(x)=p_1(x)p_2(x)\cdots p_r(x), as in (6).

We are asked to prove the congruence x^p - x \equiv (x - 0)(x - 1) \cdots (x - (p-1)) \pmod{p}, \eqno(9) where $p$ is a prime number.

The solution addresses both parts of the exercise.

I have carefully traced the construction and identified why the previous implementation produces incorrect matrices.

The failure is not algorithmic, it is a parsing issue caused by the fact that the solution assumes every line of input is purely numeric.

The solution addresses both parts of the exercise.

Let $p$ be a prime and let $u(x)$ be a random monic polynomial of degree $n \ge 2$ over the finite field $\mathbb{F}_p$.

The exercise asks for a direct algebraic simplification of two displayed identities involving content and primitive part of polynomials over a unique factorization domain $S$.

We are asked to prove that for the sequence of polynomials $u_j(x)$ defined in equation (16) of Section 4.

The solution addresses both parts of the exercise.

Let $u_0(x),u_1(x),\dots,u_{k+1}(x)$ be the Sturm sequence generated from a real polynomial $u(x)$ of degree $m=\deg(u)$ as in (29): u_0(x) = u(x),\qquad u_1(x) = u'(x), and for $j \ge 1$,

We treat the problem in two parts: first the general existence theorem for greatest common right divisors of integer matrices, then the explicit computation for the specific matrices.

We are asked to analyze **Algorithm C** for computing the greatest common divisor (gcd) of two integer polynomials of degree $n$ with coefficients bounded in absolute value by $N$, and to prove that i...

The game is played on a tree, which is an undirected, connected, acyclic graph.

The previous write-up fails at a foundational level because it treats the task as an array-sorting problem, while the exercise is about the structure of the free associative algebra over an alphabet.

Let N(d_1,\ldots,d_n;S_1,\ldots,S_n) = |S_1|\cdots |S_n| -

Let $A=(a_{ij})$ be an $n\times n$ real matrix, and let $r_i=(a_{i1},a_{i2},\ldots,a_{in})$ denote its $i$th row.

**Corrected Solution to Exercise 4.

The solution does not correctly address the statement being proved, and it does not provide a valid argument that the pseudo-remainder must be divisible by the leading coefficient $l(v)$.

The reviewer feedback does not match the exercise being solved.

The question refers to the row-naming convention of Table 1 in §4.

Let $S$ be a unique factorization domain, and let $S[x]$ denote the ring of polynomials in one indeterminate $x$ with coefficients in $S$.

Let $f(x)$ be a polynomial with integer coefficients, and suppose that $f(x)$ is irreducible over the domain of integers.

**Solution to Exercise 4.

Let $f(x)$ be a unit in the polynomial ring over a unique factorization domain $S$.

Let $S = \mathbb{F}_p$.

We are asked whether the _binary gcd algorithm_ (Algorithm 4.

**Problem 2.

Let $F=\mathbf F_p$.

We are asked to compute the pseudo-quotient $q(x)$ and pseudo-remainder $r(x)$ over the integers for u(x) = x^6 + x^5 - x^4 + 2x^3 + 3x^2 - x + 2, \qquad v(x) = 2x^2 + 2x^2 - x + 3.

We are asked to recover a literary quotation x = x_1 x_2 represented in ASCII, from the ciphertext

**Exercise 4.

Let G=\langle a\rangle=(\mathbb Z/p\mathbb Z)^\times, where $p$ is prime and $a$ is a primitive root modulo $p$.

We are asked to solve the congruence x^2 - ay^2 \equiv b \pmod{n} for integers $x$ and $y$, given that $a, b \perp n$ and $n$ is odd, without knowledge of the factorization of $n$.

**Corrected Solution to Exercise 4.

Let m=pq be a Blum integer, with

**Solution.

The requested solution is a standalone writeup, so I am providing it in a writing block.

We are asked to consider an abstract computer that can perform the operations $x + y$, $x - y$, $x \cdot y$, and $\lfloor x/y \rfloor$ on integers $x$ and $y$ of arbitrary length in one unit of time,...

The reviewer’s objections are correct.

Let $N = pq$ where $p \equiv 3 \pmod 8$ and $q \equiv 7 \pmod 8$.

We are asked to find a long chain of *successive primes*, where a prime \(q\) is a successor of a prime \(p\) if \[ q = 2^k p + 1 \] for some integer \(k \ge 0\), and both \(p\) and \(q\) are prime.

Let $N = pq$ where $p \equiv 3 \pmod 8$ and $q \equiv 7 \pmod 8$.

Equation (22) has the form T(m)\asymp m+\frac{\ln N}{\ln m}, up to multiplicative factors that vary only slowly with $m$.

**Statement.

Let $N = pq$ be the product of two distinct primes, as in the RSA scheme.

We are asked to use exercise 1.

Let $N$ be an odd positive integer with prime factorization N = q_1^{f_1} \cdots q_d^{f_d}, where the $q_i$ are distinct primes and $f_i \ge 1$.

Suppose RSA uses public exponent $e=3$.

Let $Q(A,B)=A^2-dB^2,$ and let $v_p(n)$ denote the exponent of the prime $p$ in $n$, with the convention that $v_p(n)=k \iff p^k\mid n,\quad p^{k+1}\nmid n.$ The quantity to be determined is $f(p,d)=\...

Let S(n)=\{\,p_1^{e_1}p_2^{e_2}\cdots p_m^{e_m}\le n : e_i\ge0\,\}.

Let N=5\cdot 2^n+1, and let

Let N=fr+1,\qquad 0<r\le f+1, and suppose that for every prime divisor $p$ of $f$ there exists an integer $x_p$ such that

The statement as printed in the exercise contains a typographical problem in the oscillatory term.

Let $p \ge 0$ be an integer and $q > 1$ an odd integer.

Let $S = {n : 1 < n \le N,\ n\ \text{odd},\ n\ \text{composite}}$.

Let $p$ be a prime number, and consider Algorithm B from Section 4.

Let $n\ge 3$ be odd, and let $p_n$ be the probability that Algorithm P declares $n$ to be prime when $n$ is actually composite.

Let $D$ be a given positive integer, and let $p$ range over odd primes.

Let $p_n$ and $p_{n-1}$ denote the two largest prime factors in a typical factorization, ordered so that $p_{n-1} \le p_n$.