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  2. Factorization - Wikipedia

    en.wikipedia.org/wiki/Factorization

    These factorizations work not only over the complex numbers, but also over any field, where either –1, 2 or –2 is a square. In a finite field , the product of two non-squares is a square; this implies that the polynomial x 4 + 1 , {\displaystyle x^{4}+1,} which is irreducible over the integers, is reducible modulo every prime number .

  3. Congruence of squares - Wikipedia

    en.wikipedia.org/wiki/Congruence_of_squares

    This corresponds to a set of y values whose product is a square number, i.e. one whose factorization has only even exponents. The products of x and y values together form a congruence of squares. This is a classic system of linear equations problem, and can be efficiently solved using Gaussian elimination as soon as the number of rows exceeds ...

  4. Square (algebra) - Wikipedia

    en.wikipedia.org/wiki/Square_(algebra)

    The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) 2 = x 2 ...

  5. Fermat's theorem on sums of two squares - Wikipedia

    en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of...

    On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 (if number squared is even) or 1 (if number squared is odd) modulo 4.

  6. Exponentiation by squaring - Wikipedia

    en.wikipedia.org/wiki/Exponentiation_by_squaring

    x 1 = x; x 2 = x 2 for i = k - 2 to 0 do if n i = 0 then x 2 = x 1 * x 2; x 1 = x 1 2 else x 1 = x 1 * x 2; x 2 = x 2 2 return x 1 The algorithm performs a fixed sequence of operations ( up to log n ): a multiplication and squaring takes place for each bit in the exponent, regardless of the bit's specific value.

  7. Completing the square - Wikipedia

    en.wikipedia.org/wiki/Completing_the_square

    As conventionally taught, completing the square consists of adding the third term, v 2 to + to get a square. There are also cases in which one can add the middle term, either 2 uv or −2 uv , to u 2 + v 2 {\displaystyle u^{2}+v^{2}} to get a square.

  8. Square number - Wikipedia

    en.wikipedia.org/wiki/Square_number

    The difference between any perfect square and its predecessor is given by the identity n 2 − (n − 1) 2 = 2n − 1.Equivalently, it is possible to count square numbers by adding together the last square, the last square's root, and the current root, that is, n 2 = (n − 1) 2 + (n − 1) + n.

  9. Quadratic equation - Wikipedia

    en.wikipedia.org/wiki/Quadratic_equation

    Because (a + 1) 2 = a, a + 1 is the unique solution of the quadratic equation x 2 + a = 0. On the other hand, the polynomial x 2 + ax + 1 is irreducible over F 4, but it splits over F 16, where it has the two roots ab and ab + a, where b is a root of x 2 + x + a in F 16. This is a special case of Artin–Schreier theory.