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  2. Collatz conjecture - Wikipedia

    en.wikipedia.org/wiki/Collatz_conjecture

    Equivalently, ⁠ 2n − 1 / 3 ⁠ ≡ 1 (mod 2) if and only if n ≡ 2 (mod 3). Conjecturally, this inverse relation forms a tree except for a 1–2 loop (the inverse of the 1–2 loop of the function f(n) revised as indicated above). Alternatively, replace the 3n + 1 with ⁠ n ′ / H(n ′) ⁠ where n ′ = 3n + 1 and H(n ′) is the ...

  3. Proof of Fermat's Last Theorem for specific exponents

    en.wikipedia.org/wiki/Proof_of_Fermat's_Last...

    Fermat sent the letters in which he mentioned the case in which n = 3 in 1636, 1640 and 1657. [31] Euler sent a letter to Goldbach on 4 August 1753 in which claimed to have a proof of the case in which n = 3. [32] Euler had a complete and pure elementary proof in 1760, but the result was not published. [33] Later, Euler's proof for n = 3 was ...

  4. Binomial theorem - Wikipedia

    en.wikipedia.org/wiki/Binomial_theorem

    In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power ⁠ (+) ⁠ expands into a polynomial with terms of the form ⁠ ⁠, where the exponents ⁠ ⁠ and ⁠ ⁠ are nonnegative integers satisfying ⁠ + = ⁠ and the coefficient ⁠ ⁠ of each term is a specific positive integer ...

  5. Fermat's Last Theorem - Wikipedia

    en.wikipedia.org/wiki/Fermat's_Last_Theorem

    Equivalent statement 1: x n + y n = z n, where integer n3, has no non-trivial solutions x, y, z ∈ Z. The equivalence is clear if n is even. If n is odd and all three of x , y , z are negative, then we can replace x , y , z with − x , − y , − z to obtain a solution in N .

  6. Continued fraction - Wikipedia

    en.wikipedia.org/wiki/Continued_fraction

    where c 1 = ⁠ 1 / a 1 ⁠, c 2 = ⁠ a 1 / a 2 ⁠, c 3 = ⁠ a 2 / a 1 a 3 ⁠, and in general c n+1 = ⁠ 1 / a n+1 c n ⁠. Second, if none of the partial denominators b i are zero we can use a similar procedure to choose another sequence {d i} to make each partial denominator a 1:

  7. Graham's number - Wikipedia

    en.wikipedia.org/wiki/Graham's_number

    ⋮ g 1 = n th tower: 3333333↑...↑3 (number of 3s is given by the n1 th tower) where the number of 3s in each successive tower is given by the tower just before it. The result of calculating the third tower is the value of n, the number of towers for g 1.

  8. Stirling's approximation - Wikipedia

    en.wikipedia.org/wiki/Stirling's_approximation

    An alternative version uses the fact that the Poisson distribution converges to a normal distribution by the Central Limit Theorem. [5]Since the Poisson distribution with parameter converges to a normal distribution with mean and variance , their density functions will be approximately the same:

  9. Gödel numbering - Wikipedia

    en.wikipedia.org/wiki/Gödel_numbering

    1 Simplified overview. 2 Gödel's encoding. Toggle Gödel's encoding subsection. 2.1 Example. 3 Lack of uniqueness. ... 1 3 5 7 9 11 13 17 19 23 ... 289 361 529 ...