Tag: equazioni
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There is no non-constant f ∈ ℤ[X]: f(x) is prime for every integer x
Suppose by contradiction that there exists a non-constant polynomial f with coefficients in ℤ such that f(x) is prime for every integer x. Then there exists a polynomial g ∈ ℤ[X] such that f(x)=xg(x)+p for every integer x where p is prime. Then we have for every integer x that: f(xp)=xpg(xp)+p=p(xg(xp)+1) it follows that for…
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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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Che forma ha il sottoinsieme dei punti di ℝ³ soddisfacenti l’equazione -Z⁴+8Z³+X²+Y²-16Z²=0 ?
Consideriamo il sottoinsieme S dei punti (X,Y,Z) di R³ soddisfacenti l’equazione -Z⁴+8Z³+X²+Y²-16Z²=0 . Che forma ha S ? Per capirlo prendiamo un punto (X,Y,Z) in S, allora: -Z⁴+ 8Z³ + X² + Y² -16Z² =0. Tale condizione diventa: X²+ Y² = Z⁴ -8Z³ + 16Z² = (Z² -4Z)² quindi posto v=Z, esiste u in [0,2pi)…
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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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Ring with 4 elements
Let R be a unit ring with 4 elements. Let us prove that R is commutative. R ={0,1,a,b}. Let x,y be in R and let us prove that xy=yx. If x=0 then obviously xy=0=yx. If x=1 then xy=y=yx. We then have to prove that ab=ba. We have a+1∈ R. So a+1=0 or a+1=b. If a+1=0…
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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.…
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The Diophantine equation X⁴-Y⁴-X=0
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…
Marco Damele 