Prime Factors of Cyclotomic Class Numbers
... To search for the prime factors of Pd, we therefore try as divisors of Pd only the numbers in the arithmetic progression 2xdx + 1 (x — 1, 2, 3, . . . ). The first such divisor is either a prime or a power of a prime. After removing all such factors below some limit, an attempt can be made to represe ...
... To search for the prime factors of Pd, we therefore try as divisors of Pd only the numbers in the arithmetic progression 2xdx + 1 (x — 1, 2, 3, . . . ). The first such divisor is either a prime or a power of a prime. After removing all such factors below some limit, an attempt can be made to represe ...
Simple Continued Fractions for Some Irrational Numbers
... repeats. Thus B(u, ∞) is irrational, and its continued fraction does not terminate. Theorem 4 The first 2v partial denominators of the continued fraction for B(u, v) are identical with those of the continued fraction for B(u, ∞). Proof. Examination of part (B) of Theorem 1 shows that the first 2v pa ...
... repeats. Thus B(u, ∞) is irrational, and its continued fraction does not terminate. Theorem 4 The first 2v partial denominators of the continued fraction for B(u, v) are identical with those of the continued fraction for B(u, ∞). Proof. Examination of part (B) of Theorem 1 shows that the first 2v pa ...
Lecture slides (full content)
... prove ¬Q => ¬P (equivalent to P => Q) Chain of implication: When is this useful? when the reverse direction is easier to prove than the original. ...
... prove ¬Q => ¬P (equivalent to P => Q) Chain of implication: When is this useful? when the reverse direction is easier to prove than the original. ...
What is. . . an L-function? - Mathematisch Instituut Leiden
... Λ(X, s) = (X)Λ(X ∗ , k − s). In the case where L(X, s) admits (or is expected to admit) such a functional equation, the vertical line {s ∈ C |
... Λ(X, s) = (X)Λ(X ∗ , k − s). In the case where L(X, s) admits (or is expected to admit) such a functional equation, the vertical line {s ∈ C |
ADDING AND COUNTING Definition 0.1. A partition of a natural
... 3, and the next one tells us that there are no congruences as in Theorem 0.8 for primes other than 5, 7, and 11. Theorem 0.10 (Radu 2011). There are no arithmetic progressions An + B such that p(An + B) ≡ 0 (mod 2) or p(An + B) ≡ 0 (mod 3). Theorem 0.11 (Ahlgren, Boylan 2005). The only (l, a) such t ...
... 3, and the next one tells us that there are no congruences as in Theorem 0.8 for primes other than 5, 7, and 11. Theorem 0.10 (Radu 2011). There are no arithmetic progressions An + B such that p(An + B) ≡ 0 (mod 2) or p(An + B) ≡ 0 (mod 3). Theorem 0.11 (Ahlgren, Boylan 2005). The only (l, a) such t ...
Lacunary recurrences for Eisenstein series
... forms. Here, we provide such a proof, and in particular show that Romik’s example is a natural, and especially symmetric, instance of general relations among products of two The second author thanks the University of Cologne and the DFG for their generous support via the University of Cologne postdo ...
... forms. Here, we provide such a proof, and in particular show that Romik’s example is a natural, and especially symmetric, instance of general relations among products of two The second author thanks the University of Cologne and the DFG for their generous support via the University of Cologne postdo ...
On the difference of consecutive primes.
... We reduce our problem to the proof of the -following theorem . THEOREM II . For a certain positive constant c2 , we can find c2 pn log pn /(loglog ontip)2scfheame,gurv relatively prime to the product p1 p 2 .. . . pn, i .e . each of these integers is divisible by at least one of the primes p 1 , p2 ...
... We reduce our problem to the proof of the -following theorem . THEOREM II . For a certain positive constant c2 , we can find c2 pn log pn /(loglog ontip)2scfheame,gurv relatively prime to the product p1 p 2 .. . . pn, i .e . each of these integers is divisible by at least one of the primes p 1 , p2 ...
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"".