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  2. Basel problem - Wikipedia

    en.wikipedia.org/wiki/Basel_problem

    The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [ 1 ] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences . [ 2 ]

  3. Hurwitz problem - Wikipedia

    en.wikipedia.org/wiki/Hurwitz_problem

    The Hurwitz problem for the field K is to find general relations of the form (+ +) (+ +) = (+ +) , with the z being bilinear forms in the x and y: that is, each z is a K-linear combination of terms of the form x i y j.

  4. Landau's problems - Wikipedia

    en.wikipedia.org/wiki/Landau's_problems

    Goldbach's weak conjecture, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of Goldbach's conjecture. Ivan Vinogradov proved it for large enough n (Vinogradov's theorem) in 1937, [1] and Harald Helfgott extended this to a full proof of Goldbach's weak conjecture in 2013. [2] [3] [4]

  5. Hilbert's seventeenth problem - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_seventeenth_problem

    A result of Albrecht Pfister [8] shows that a positive semidefinite form in n variables can be expressed as a sum of 2 n squares. [9] Dubois showed in 1967 that the answer is negative in general for ordered fields. [10] In this case one can say that a positive polynomial is a sum of weighted squares of rational functions with positive ...

  6. Sum of squares - Wikipedia

    en.wikipedia.org/wiki/Sum_of_squares

    Legendre's three-square theorem states which numbers can be expressed as the sum of three squares; Jacobi's four-square theorem gives the number of ways that a number can be represented as the sum of four squares. For the number of representations of a positive integer as a sum of squares of k integers, see Sum of squares function.

  7. Sum of two squares theorem - Wikipedia

    en.wikipedia.org/wiki/Sum_of_two_squares_theorem

    Therefore, the theorem states that it is expressible as the sum of two squares. Indeed, 2450 = 7 2 + 49 2. The prime decomposition of the number 3430 is 2 · 5 · 7 3. This time, the exponent of 7 in the decomposition is 3, an odd number. So 3430 cannot be written as the sum of two squares.

  8. Mathematical proof - Wikipedia

    en.wikipedia.org/wiki/Mathematical_proof

    In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually infinitely many) cases are provable. [ 15 ]

  9. Mathematical induction - Wikipedia

    en.wikipedia.org/wiki/Mathematical_induction

    Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. [1] [2]Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold.