- ESAIM: Proceedings
... From technical point of view, it is more convenient to connect size-biased Galton–Watson trees with Galton–Watson branching processes with immigration, described as follows. A Galton–Watson branching processes with immigration starts with no individual (say), and is characterized by a reproduction l ...
... From technical point of view, it is more convenient to connect size-biased Galton–Watson trees with Galton–Watson branching processes with immigration, described as follows. A Galton–Watson branching processes with immigration starts with no individual (say), and is characterized by a reproduction l ...
On values taken by the largest prime factor of shifted primes
... 2.2. Necessary tools Our principal tool is the following result, which follows immediately from the Bombieri-Vinogradov theorem (see [9]) in the range 0 < # < 1=2, from [2, Theorem 1] in the range 1=2 ≤ # ≤ 13=25, and from the main theorem of [23] in the range 13=25 < # < 17=32. LEMMA 2.1. There exi ...
... 2.2. Necessary tools Our principal tool is the following result, which follows immediately from the Bombieri-Vinogradov theorem (see [9]) in the range 0 < # < 1=2, from [2, Theorem 1] in the range 1=2 ≤ # ≤ 13=25, and from the main theorem of [23] in the range 13=25 < # < 17=32. LEMMA 2.1. There exi ...
Congruent Numbers Via the Pell Equation and its Analogous
... En (Q) : y 2 = x3 − n2 x contains a rational point with y 6= 0, equivalently, a rational point of infinite order [13]. In 1929, Nagell [18] had a very short and elementary proof of the fact that the rank of En (Q) is zero in the case of n = p ≡ 3(mod8) for a prime number p. Thus these numbers are no ...
... En (Q) : y 2 = x3 − n2 x contains a rational point with y 6= 0, equivalently, a rational point of infinite order [13]. In 1929, Nagell [18] had a very short and elementary proof of the fact that the rank of En (Q) is zero in the case of n = p ≡ 3(mod8) for a prime number p. Thus these numbers are no ...
DISTRIBUTION OF RESIDUES MODULO p - Harish
... Also, γ(N ) = 1 for all N ≥ 2. Proof. First we shall prove that Γ(2) ≤ 9. It is enough to prove that r(2, p) ≤ 9 for every prime p ≥ 11. If 10 is a quadratic non-residue mod p, then either 2 or 5 are quadratic residue mod p. Hence (1, 2) or (4, 5) are pairs of quadratic residues mod p. If 10 is a qu ...
... Also, γ(N ) = 1 for all N ≥ 2. Proof. First we shall prove that Γ(2) ≤ 9. It is enough to prove that r(2, p) ≤ 9 for every prime p ≥ 11. If 10 is a quadratic non-residue mod p, then either 2 or 5 are quadratic residue mod p. Hence (1, 2) or (4, 5) are pairs of quadratic residues mod p. If 10 is a qu ...
Number Theory and Combinatorics
... table such that for each i, 1 ≤ i ≤ 9 the product of the numbers in row i equals the product of the numbers in column i. Solution The key observation is the following: if row k contains a prime number p > 40, then the same number must be contained by column k, as well. Therefore, all prime numbers f ...
... table such that for each i, 1 ≤ i ≤ 9 the product of the numbers in row i equals the product of the numbers in column i. Solution The key observation is the following: if row k contains a prime number p > 40, then the same number must be contained by column k, as well. Therefore, all prime numbers f ...
On Divisibility By Nine of the Sums
... the theorem is described and six new exceptions to Gardner's rule are found. ...
... the theorem is described and six new exceptions to Gardner's rule are found. ...
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"".