Question

Find all monic polynomials $f(x)$ having integer coefficients such that $f(0)=2020$ and for any irrational $x$, $f(x)$ is also irrational.

score 450 · Answer 1 · 2020-07-17 05:51:07

450

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Find all monic polynomials $f(x)$ having integer coefficients such that $f(0)=2020$ and for any irrational $x$, $f(x)$ is also irrational.

Published on 17 Jul 2020 - 5:51

score 135 · Answer 2 · 2026-03-29 10:09:45

135

Views

How is the "diagonal proof" valid?

Published on 29 Mar 2026 - 10:09

score 106 · Answer 3 · 2026-03-26 17:51:58

106

Views

Which irrationals become rational for some positive integer power?

Published on 26 Mar 2026 - 17:51

score 522 · Answer 4 · 2026-02-23 06:39:39

522

Views

Prove that the sum $ \sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$ is irrational

Published on 23 Feb 2026 - 6:39

score 117 · Answer 5 · 2020-07-24 09:38:50

117

Views

Number of irrational coordinates of the circle $x^2+y^2=5$.

Published on 24 Jul 2020 - 9:38

score 47 · Answer 6 · 2020-07-25 20:04:10

47

Views

How to express any irrational number in radicals as a trigonometric expression

Published on 25 Jul 2020 - 20:04

score 174 · Answer 7 · 2020-07-31 05:10:56

174

Views

INMO : Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers

Published on 31 Jul 2020 - 5:10

score 218 · Answer 8 · 2026-03-26 21:27:25

218

Views

Quadratic irrationals with continued fraction of period one

Published on 26 Mar 2026 - 21:27

score 20 · Answer 9 · 2016-07-07 12:48:59

20

Views

Irrationality of a real number $b$ that is related to a real number $a$ by the umbral Taylor series.

Published on 07 Jul 2016 - 12:48

score 94 · Answer 10 · 2016-07-07 17:39:11

94

Views

Can an irrational number be expressed as a sum of other irrational numbers, at least one of which is not an integral multiple of the required number?

Published on 07 Jul 2016 - 17:39

score 102 · Answer 11 · 2016-07-09 00:02:25

102

Views

What is the term for 'PI-indexing'?

Published on 09 Jul 2016 - 0:02

score 400 · Answer 12 · 2026-03-28 00:28:44

400

Views

Proof of $\pi^e$ and $e^\pi$ Being Irrational

Published on 28 Mar 2026 - 0:28

score 441 · Answer 13 · 2026-02-23 05:31:45

441

Views

Greedy algorithm Egyptian fractions for irrational numbers - patterns and irrationality proofs

Published on 23 Feb 2026 - 5:31

score 203 · Answer 14 · 2026-03-29 06:00:37

203

Views

Notation for representing ANY number?

Published on 29 Mar 2026 - 6:00

score 218 · Answer 15 · 2026-03-28 00:36:41

218

Views

$\pi$ and $e$ as coded trajectories

Published on 28 Mar 2026 - 0:36

List Question

Find all monic polynomials $f(x)$ having integer coefficients such that $f(0)=2020$ and for any irrational $x$, $f(x)$ is also irrational.

How is the "diagonal proof" valid?

Which irrationals become rational for some positive integer power?

Prove that the sum $ \sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$ is irrational

Number of irrational coordinates of the circle $x^2+y^2=5$.

How to express any irrational number in radicals as a trigonometric expression

INMO : Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers

Quadratic irrationals with continued fraction of period one

Irrationality of a real number $b$ that is related to a real number $a$ by the umbral Taylor series.

Can an irrational number be expressed as a sum of other irrational numbers, at least one of which is not an integral multiple of the required number?

What is the term for 'PI-indexing'?

Proof of $\pi^e$ and $e^\pi$ Being Irrational

Greedy algorithm Egyptian fractions for irrational numbers - patterns and irrationality proofs

Notation for representing ANY number?

$\pi$ and $e$ as coded trajectories

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