3-6 Fundamental Theorem of Algebra Day 1
... Write the simplest polynomial function with the given zeros. ...
... Write the simplest polynomial function with the given zeros. ...
Oliver Johnson and Christina Goldschmidt 1. Introduction
... For the first two examples he considers, this is relatively straightforward. That is, in Corollary 4.5.1 he uses the fact that the binomial coefficients nk have generating function (1 + x)n , and in Corollary 4.5.2 he uses the fact that the Stirling numbers of the first kind have generating function ...
... For the first two examples he considers, this is relatively straightforward. That is, in Corollary 4.5.1 he uses the fact that the binomial coefficients nk have generating function (1 + x)n , and in Corollary 4.5.2 he uses the fact that the Stirling numbers of the first kind have generating function ...
M3P14 Elementary Number Theory—Problem Sheet 4.
... Try to find a pattern and make a guess as to exactly which primes have this form. Use the technique of Fermat descent to prove that your guess is correct. The only places that you are likely to get stuck is that at one stage you might find a prime p such that 2p = a2 + 2b2 and you will want to deduc ...
... Try to find a pattern and make a guess as to exactly which primes have this form. Use the technique of Fermat descent to prove that your guess is correct. The only places that you are likely to get stuck is that at one stage you might find a prime p such that 2p = a2 + 2b2 and you will want to deduc ...
3. CATALAN NUMBERS Corollary 1. cn = 1
... Proposition 2. The number of triangulations of a convex (n + 2)-gon by diagonals is cn . To count triangulations correctly one should have the vertices of the polygon numbered. For example, for a square with the vertices 1, 2, 3, 4 there are c2 = 2 triangulations: a diagonal 1; 3 or a diagonal 2; 4. ...
... Proposition 2. The number of triangulations of a convex (n + 2)-gon by diagonals is cn . To count triangulations correctly one should have the vertices of the polygon numbered. For example, for a square with the vertices 1, 2, 3, 4 there are c2 = 2 triangulations: a diagonal 1; 3 or a diagonal 2; 4. ...
A COMBINATORIAL PROOF OF A RESULT FROM NUMBER
... Geometrically, rk (n) counts the number of points with integer coordinates on the kdimensional sphere x21 + x22 + · · · + x2k = n. Similarly, 2k tk (n) counts the number of points with integer coordinates on the k-dimensional sphere (x1 + 12 )2 +(x2 + 12 )2 +· · ·+(xk + 12 )2 = 2n + k4 . A great dea ...
... Geometrically, rk (n) counts the number of points with integer coordinates on the kdimensional sphere x21 + x22 + · · · + x2k = n. Similarly, 2k tk (n) counts the number of points with integer coordinates on the k-dimensional sphere (x1 + 12 )2 +(x2 + 12 )2 +· · ·+(xk + 12 )2 = 2n + k4 . A great dea ...
Prime Numbers and Irreducible Polynomials
... to produce prime numbers from irreducible polynomials. This conjecture is still one of the major unsolved problems in number theory when the degree of f is greater than one. When f is linear, the conjecture is true, of course, and follows from Dirichlet’s theorem on primes in arithmetic progressions ...
... to produce prime numbers from irreducible polynomials. This conjecture is still one of the major unsolved problems in number theory when the degree of f is greater than one. When f is linear, the conjecture is true, of course, and follows from Dirichlet’s theorem on primes in arithmetic progressions ...
A coprimality condition on consecutive values of polynomials
... Later, the combined efforts of Pillai [14] and Brauer [3] gave a more explicit result, namely that gs = Gs = 17. We note that interest in such a problem is twofold. On one hand, Pillai aimed at the solution of the classical Diophantine problem whether the product of consecutive integers can be a per ...
... Later, the combined efforts of Pillai [14] and Brauer [3] gave a more explicit result, namely that gs = Gs = 17. We note that interest in such a problem is twofold. On one hand, Pillai aimed at the solution of the classical Diophantine problem whether the product of consecutive integers can be a per ...
Acta Mathematica et Informatica Universitatis Ostraviensis - DML-CZ
... have at most three nontrivial solutions, and suggested that such systems have at most one nontrivial solution, provided that they are not of a very specific form which is described in [3]. ...
... have at most three nontrivial solutions, and suggested that such systems have at most one nontrivial solution, provided that they are not of a very specific form which is described in [3]. ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".