DENSITY AND SUBSTANCE
... it justice, as the proof is nowhere near as elementary. In fact, for decades after Erdős and Turán made this conjecture, it was one of the great unsolved problems in number theory. Endre Szemerédi unlocked the mystery with a combinatorial proof nearly 40 years after the original conjecture. As mo ...
... it justice, as the proof is nowhere near as elementary. In fact, for decades after Erdős and Turán made this conjecture, it was one of the great unsolved problems in number theory. Endre Szemerédi unlocked the mystery with a combinatorial proof nearly 40 years after the original conjecture. As mo ...
DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION
... If x = p/q ∈ Q, let us take a sequence {xn } of irrational numbers such that xn → x when n → ∞; then f (xn ) = 0 for every n and the sequence {f (xn )} does not converge to f (x) = 1/q, so f is not continuous at x. On the other hand, for x ∈ R \ Q, let us see that f is continuous at x by checking th ...
... If x = p/q ∈ Q, let us take a sequence {xn } of irrational numbers such that xn → x when n → ∞; then f (xn ) = 0 for every n and the sequence {f (xn )} does not converge to f (x) = 1/q, so f is not continuous at x. On the other hand, for x ∈ R \ Q, let us see that f is continuous at x by checking th ...
Binomial coefficients and p-adic limits
... bk k! and it is obvious that the only possible primes in the denominator are prime factors of b or a prime factor of k!. It is not obvious why every prime factor of k! that is not a factor of b gets completely canceled out when the ratio in kr is simplified. We will explain this purely algebraic ph ...
... bk k! and it is obvious that the only possible primes in the denominator are prime factors of b or a prime factor of k!. It is not obvious why every prime factor of k! that is not a factor of b gets completely canceled out when the ratio in kr is simplified. We will explain this purely algebraic ph ...
2E Numbers and Sets What is an equivalence relation on a set X? If
... (a) What is the highest common factor of two positive integers a and b? Show that the highest common factor may always be expressed in the form λa + µb, where λ and µ are integers. (a) The hcf of a and b is a positive integer c such that c | a and c | b and such that if d | a and d | b then d | c (c ...
... (a) What is the highest common factor of two positive integers a and b? Show that the highest common factor may always be expressed in the form λa + µb, where λ and µ are integers. (a) The hcf of a and b is a positive integer c such that c | a and c | b and such that if d | a and d | b then d | c (c ...
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"".