Discrete Mathematics
... proofs. Merely stating the facts, without saying something about why these facts are valid, would be terribly far from the spirit of mathematics and would make it impossible to give any idea about how it works. Thus, wherever possible, we’ll give the proofs of the theorems we state. Sometimes this i ...
... proofs. Merely stating the facts, without saying something about why these facts are valid, would be terribly far from the spirit of mathematics and would make it impossible to give any idea about how it works. Thus, wherever possible, we’ll give the proofs of the theorems we state. Sometimes this i ...
Elementary Evaluation of Convolution Sums
... of representations of a positive integer by the octonary quadratic forms Equation 1.4 and Equation 1.5. Then we determine explicit formulae for the number of representations of a positive integer n by the octonary quadratic forms Equation 1.4 and Equation 1.5, whenever αβ has the above form and is s ...
... of representations of a positive integer by the octonary quadratic forms Equation 1.4 and Equation 1.5. Then we determine explicit formulae for the number of representations of a positive integer n by the octonary quadratic forms Equation 1.4 and Equation 1.5, whenever αβ has the above form and is s ...
Pythagorean Triples Solution Commentary:
... latter square is removed, is it possible to transform the remaining “area” into a square of side length b? You might want to have students explore Schwaller’s (1979) interesting approach to this question, which ends up generating Pythagorean triples. In Problem #1, Pythagoras’ method produced some P ...
... latter square is removed, is it possible to transform the remaining “area” into a square of side length b? You might want to have students explore Schwaller’s (1979) interesting approach to this question, which ends up generating Pythagorean triples. In Problem #1, Pythagoras’ method produced some P ...
fundamental concepts of algebra - Department of Mathematical
... Of course, thinking of 0 and −a in terms of addition leads to some interesting questions. Why is 0 times any number equal to 0? Why is the product of two negative numbers a positive number? More generally, what does multiplication by a negative number really mean? Is it still repeated addition? We w ...
... Of course, thinking of 0 and −a in terms of addition leads to some interesting questions. Why is 0 times any number equal to 0? Why is the product of two negative numbers a positive number? More generally, what does multiplication by a negative number really mean? Is it still repeated addition? We w ...
Solution
... Substituting back in equation (4) we obtain that y = 10 − 8 · k, even if we don’t need the values for y. Thus the solutions of 45x ≡ 15 (mod 24) are x = −5 + 8 · k, where k is any integer. The solution set consists of the numbers {. . . , −21, −13, −5, 3, 11, 19, 27, . . . }. Observe, tough, that th ...
... Substituting back in equation (4) we obtain that y = 10 − 8 · k, even if we don’t need the values for y. Thus the solutions of 45x ≡ 15 (mod 24) are x = −5 + 8 · k, where k is any integer. The solution set consists of the numbers {. . . , −21, −13, −5, 3, 11, 19, 27, . . . }. Observe, tough, that th ...
From Apollonian Circle Packings to Fibonacci Numbers
... (1) For each n 1 there are finitely many primitive integral Apollonian circle packings having a root quadruple with smallest element equal to n. (2) The number of such packings Nroot( n) is given by the (Legendre) class number counting primitive binary quadratic form classes of discriminant 4n2 unde ...
... (1) For each n 1 there are finitely many primitive integral Apollonian circle packings having a root quadruple with smallest element equal to n. (2) The number of such packings Nroot( n) is given by the (Legendre) class number counting primitive binary quadratic form classes of discriminant 4n2 unde ...
Project Gutenberg`s Diophantine Analysis, by Robert Carmichael
... entirely, since that subject would require a volume in itself. The reader will therefore miss such an elegant theorem as the following: Every positive integer may be represented as the sum of four squares. Some methods of frequent use in the theory of quadratic forms, in particular that of continued ...
... entirely, since that subject would require a volume in itself. The reader will therefore miss such an elegant theorem as the following: Every positive integer may be represented as the sum of four squares. Some methods of frequent use in the theory of quadratic forms, in particular that of continued ...
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"".