Marco Damele

Category: Un pò di matematica

  • Finite groups of class 2 e 3.

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    Marco Damele
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    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…

  • The Diophantine equation 4X⁴-16X³+20X²-8X-3=Y²

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    Marco Damele
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    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.…

  • The Diophantine equation X⁴-Y⁴-X=0

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    Marco Damele
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    Let’s solve the Diophantine equation X⁴-Y⁴-X=0 together so as to see a standard approach that is encountered in solving Diophantine equations of the form Z(f(Z)-1)=Y² since f is a function of Z. For any typos I invite you to report them in the comments. Let’s suppose that (X,Y) is a non-trivial solution, that is, X…

  • A simple Diophantine equation

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    Marco Damele
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    Let’s solve a Diophantine equation together using only elementary considerations. Suppose we are asked to determine all the integers X and Y such that X³-X+1=3Y⁵. As is usually done, we assume that such a solution exists, let’s say (X,Y), and we try to understand if we can make the value of X and Y explicit…

  • Valore esatto di cos(2π/5)

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    Marco Damele
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    Exercise for high school students who have studied complexes and are wondering what they can be used for. Let’s show that cos(2π/5)=(-1+√5)/4. (1) Let w=cos(2π/5)+isin(2π/5) and x=Re(w). We want to show that x= (-1+√5)/4. Let’s observe that: x=(w+w⁻¹)/2 (2) where w⁻¹ is the multiplicative inverse of w. To see this, let’s first recall that if…

  • Complex roots

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    Given a complex w and a positive natural number n, we say that z is the nth root of w if zⁿ=w. We set: ⁿ√w = { z ∈ ℂ : zⁿ=w} that is, ⁿ√w is the set of n-th roots of w. From the fundamental theorem of algebra there are exactly n n-th roots…

  • A Diophantine equation

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    Marco Damele
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    Let us determine all the prime positive p and q such that p²-2q²=1. We observe that p=3 and q=2 satisfy the question since they are both positive primes and (3)²-2(2)²=9-8=1. Conversely, if p and q are prime positives such that p²-2q²=1 then p²=2q²+1 therefore p² is odd or p is odd let’s say p=2k+1 then…

  • 0 varietà topologiche

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    CLASSIFICAZIONE DELLE 0-VARIETÀ TOPOLOGICHE Ricordiamo che una varietà topologica di dimensione n∈ℕ (detta anche n-varietà topologica) è uno spazio topologico (M,τ) tale che sia: 1) N2 (cioè che ammetta una base numerabile ovvero che esista B⊆τ numerabile tale che ∀ V∈τ V è unione di elementi di B )2) T2 (cioè che ∀x,y ∈ M…

  • A special equation in a Boolean ring

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    Marco Damele
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    Let us consider a ring R such that ∀ x ∈ R x²=x (such a ring is called Boolean). Let us solve the equation in R x R x R: (XY)⁶-(YX)⁶=2X⁷-4Y⁵+8YZ³ Let us just recall that a ring R is a set provided with two binary operations + and • such that (R,+) is an…

  • If ℤⁿ≅ℤᵐ then n=m.

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    Marco Damele
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    Let ψ:ℤⁿ—>ℤᵐ a group isomorphism. Let’s prove that n=m.Let: H=2ℤⁿ={(2z1,…,2zn) : zi∈ℤ ∀ i=1,..,n} ≤ ℤⁿK= 2ℤᵐ={(2z1,…,2zm) : zi∈ℤ ∀ i=1,..,m} ≤ ℤᵐ Observe that ψ(H)=K. Indeed if (2z1,…,2zn) ∈ H then ψ (2z1,…,2zn) = 2ψ(z1,…,zn). Viceversa if (2z1,…,2zm) ∈ K then ψ⁻¹(2z1,…,2zm)∈H —> (2z1,…,2zm) ∈ψ(H). Now, since ℤᵐ e ℤⁿ are abelian H and…