Tag: teoria dei numeri
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Six proofs that √2 is irrational
Dimostriamo che √2 è irrazionale usando 6 dimostrazioni diverse. 1° Proof: Suppose that √2 ∈ ℚ then there exist two coprime integers a and b such that √2=a/b. It follows that 2b²=a². So a² is even, that is, a is even, or a=2k with k being an integer. But then 2b²=4k², or b²=2k², from which…
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Every year admits a Friday the 13th
Theorem: Every year admits a Friday the 13th Proof: Note that a year admits a Friday the 13th if there is a month that begins with Sunday. We denote the days of the week with numbers. Specifically, 0 denotes Sunday, 1 Monday, up to 6, a number that denotes Saturday. If k is a natural…
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n-agonal numbers
Numbers are awesome. Here’s why. Yesterday I made a video about the so-called “n-gonal numbers”. I’ll explain what they are as we go. Let’s start by saying that a natural number k is said to be “square” if k=n² for some natural number n. This definition, which is now completely algebraic, was given in the…
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A Diophantine sistem.
We want to determine all integers x₁,…,x₂₀₁₁ such that If such integers exist, by Fermat’s little theorem we would have that xi⁷=xi (mod 7) for every i=1,…,2011. So 123=321 (mod 7) or 4=6 (mod 7) which is absurd.
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If K is a field (K,+) it is not isomorphic to (K{0},•).
Let (K,+,•) be a field, let us prove that its additive group (K,+) is not isomorphic to the group of invertibles of K, that is (K{0},•). Let us suppose that there is an isomorphism between them, then, since isomorphisms preserve the order of an element, we would have that: |{x ∈ K : x+x=0}|=|{x∈ K{0}…
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Can a polynomial of degree n have more than n roots?
Let’s start with some definitions. Let R be a ring and f∈R [X] a polynomial of degree n≥0, then f=a0+a1X+…+anXⁿ where a0,a1,…,an are elements of R. An element α∈R is called a root of f if f(α)=0, that is, if a0+a1α+…+anαⁿ=0. For example, the polynomial X²-2 ∈ ℝ[X] has as its only roots √2 and…
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Finite groups of class 2 e 3.
Let G be a finite group. We want to classify all finite groups with exactly two and three conjugacy classes. The number of conjugacy classes of G is called the class of G and is denoted by Cl(G). In the following if x ∈ G the conjugacy class of x is denoted by Cl(x). Then…
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The Diophantine equation 4X⁴-16X³+20X²-8X-3=Y²
Let’s solve the Diophantine equation: 4X⁴-16X³+20X²-8X-3=Y². Let’s suppose that X,Y are two integers satisfying the equation then: (2X²-4X-1)(2X²-4X+3)=Y². Let’s observe that 2X²-4X-1 and 2X²-4X+3 are coprime since if p is a possible common prime divisor it must be odd, let’s say it is p and we would have that -4=0 mod p therefore p=2, absurd.…
Marco Damele 