Question

Let $f(x)=x^3+x^2-5$. Show that for $n=1, 2,3, ...$ there is a unique $x_n$ modulo $7^n$ such that $f(x_n)\equiv 0\pmod{7^n}$.

score 75 · Answer 1 · 2026-02-23 10:06:16

75

Views

Let $f(x)=x^3+x^2-5$. Show that for $n=1, 2,3, ...$ there is a unique $x_n$ modulo $7^n$ such that $f(x_n)\equiv 0\pmod{7^n}$.

Published on 23 Feb 2026 - 10:06

score 1.1k · Answer 2 · 2026-03-22 12:27:09

1.1k

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solve x for a cubic congruence equation with large prime mod.

Published on 22 Mar 2026 - 12:27

score 242 · Answer 3 · 2026-03-22 14:30:12

242

Views

rewriting a polynomial congruent to 0 mod m

Published on 22 Mar 2026 - 14:30

score 34 · Answer 4 · 2026-03-22 14:59:25

34

Views

$f \in\mathbb Q[X]$, $ [f] = [f \operatorname{div} X^5] $ in $ \mathbb Q[X]/\equiv_{X^5} $

Published on 22 Mar 2026 - 14:59

score 220 · Answer 5 · 2026-03-22 10:06:39

220

Views

How to solve system of congruences of polynomial?

Published on 22 Mar 2026 - 10:06

score 68 · Answer 6 · 2026-02-23 10:06:16

68

Views

if a homogeneous polynomial is the square of a polynomial modulo $x^2 + y^2 - 1$, is it also congruent to a sum of squares of homogeneous polynomials?

Published on 23 Feb 2026 - 10:06

score 34 · Answer 7 · 2026-02-23 09:59:56

34

Views

Polynomial Congruence Equation

Published on 23 Feb 2026 - 9:59

score 50 · Answer 8 · 2026-02-23 10:04:43

50

Views

corollary of the partition congruence

Published on 23 Feb 2026 - 10:04

score 65 · Answer 9 · 2026-02-23 10:07:18

65

Views

Is there any prime $p$ such that $6x^3 − p^2 − y^2 = 0$ has an integer solution?

Published on 23 Feb 2026 - 10:07

score 33 · Answer 10 · 2026-03-22 09:17:53

33

Views

The number of solutions to the congruence equation $P(x) \equiv 0 (\mathrm{mod}\ p^\alpha)$

Published on 22 Mar 2026 - 9:17

score 82 · Answer 11 · 2026-04-06 02:58:59

82

Views

Polynomials in Finite Field

Published on 06 Apr 2026 - 2:58

score 552 · Answer 12 · 2026-03-22 15:31:55

552

Views

Solving the congruence $3x^2 + 6x + 1 \equiv 0 \pmod {19}$

Published on 22 Mar 2026 - 15:31

score 323 · Answer 13 · 2026-03-21 18:10:08

323

Views

Can the concept of congruence be applied to the remainder of a polynomial division?

Published on 21 Mar 2026 - 18:10

score 64 · Answer 14 · 2026-04-05 22:06:37

64

Views

Understanding the following lemma

Published on 05 Apr 2026 - 22:06

score 156 · Answer 15 · 2026-02-23 10:02:04

156

Views

Is there a solution for $x^6 \equiv 5 \pmod {71}$?

Published on 23 Feb 2026 - 10:02

List Question

Let $f(x)=x^3+x^2-5$. Show that for $n=1, 2,3, ...$ there is a unique $x_n$ modulo $7^n$ such that $f(x_n)\equiv 0\pmod{7^n}$.

solve x for a cubic congruence equation with large prime mod.

rewriting a polynomial congruent to 0 mod m

$f \in\mathbb Q[X]$, $ [f] = [f \operatorname{div} X^5] $ in $ \mathbb Q[X]/\equiv_{X^5} $

How to solve system of congruences of polynomial?

if a homogeneous polynomial is the square of a polynomial modulo $x^2 + y^2 - 1$, is it also congruent to a sum of squares of homogeneous polynomials?

Polynomial Congruence Equation

corollary of the partition congruence

Is there any prime $p$ such that $6x^3 − p^2 − y^2 = 0$ has an integer solution?

The number of solutions to the congruence equation $P(x) \equiv 0 (\mathrm{mod}\ p^\alpha)$

Polynomials in Finite Field

Solving the congruence $3x^2 + 6x + 1 \equiv 0 \pmod {19}$

Can the concept of congruence be applied to the remainder of a polynomial division?

Understanding the following lemma

Is there a solution for $x^6 \equiv 5 \pmod {71}$?

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