the origins of the genus concept in quadratic forms
... first results is a connection between quadratic residues and the representation problem for general quadratic forms: THEOREM 3: Let m be a natural number that is represented by the form ax 2 bxy cy 2 . Then b 2 4ac is a quadratic residue modulo m. One of Lagrange’s primary innovations was ...
... first results is a connection between quadratic residues and the representation problem for general quadratic forms: THEOREM 3: Let m be a natural number that is represented by the form ax 2 bxy cy 2 . Then b 2 4ac is a quadratic residue modulo m. One of Lagrange’s primary innovations was ...
Prime Factorization
... A prime number is a whole number greater than 1 that has exactly two factors, 1 and itself. A composite number is a whole number greater than 1 that has more than two factors. Every composite number can be written as the product of prime numbers in exactly one way if you ignore the order of the fact ...
... A prime number is a whole number greater than 1 that has exactly two factors, 1 and itself. A composite number is a whole number greater than 1 that has more than two factors. Every composite number can be written as the product of prime numbers in exactly one way if you ignore the order of the fact ...
The Goldston-Pintz-Yıldırım sieve and some applications
... conjecture. The twin prime conjecture asserts that the gap between consecutive primes is ifinitely often as small as it possibly can be, that is, pn+1 − pn = 2 for infinitely many n. Goldston-Pintz-Yıldırım[15] were able to use their method to prove, conditionally, that pn+1 −pn 6 16 for infinitely ...
... conjecture. The twin prime conjecture asserts that the gap between consecutive primes is ifinitely often as small as it possibly can be, that is, pn+1 − pn = 2 for infinitely many n. Goldston-Pintz-Yıldırım[15] were able to use their method to prove, conditionally, that pn+1 −pn 6 16 for infinitely ...
Universal quadratic forms and the 290-Theorem
... to Legendre’s three squares form x2 + y 2 + z 2 . This 3-dimensional lattice L3 is well-known to be unique in its genus, and it represents all positive integers not of the form 4a (8k + 7) for some integer a. Suppose [m] is the Gram matrix of the orthogonal complement of L3 in L4 . We wish to show t ...
... to Legendre’s three squares form x2 + y 2 + z 2 . This 3-dimensional lattice L3 is well-known to be unique in its genus, and it represents all positive integers not of the form 4a (8k + 7) for some integer a. Suppose [m] is the Gram matrix of the orthogonal complement of L3 in L4 . We wish to show t ...
Solutions - CMU Math
... First of all, we know that k! ≡ 0 (mod 9) for all k ≥ 6. Thus, we only need to find (1! + 2! + 3! + 4! + 5!) (mod 9). 1! ≡ 1 (mod 9) 2! ≡ 2 (mod 9) 3! ≡ 6 (mod 9) 4! ≡ 24 ≡ 6 (mod 9) 5! ≡ 5 · 6 ≡ 30 ≡ 3 (mod 9) Thus, (1! + 2! + 3! + 4! + 5!) ≡ 1 + 2 + 6 + 6 + 3 ≡ 18 ≡ 0 (mod 9) so the remainder is 0 ...
... First of all, we know that k! ≡ 0 (mod 9) for all k ≥ 6. Thus, we only need to find (1! + 2! + 3! + 4! + 5!) (mod 9). 1! ≡ 1 (mod 9) 2! ≡ 2 (mod 9) 3! ≡ 6 (mod 9) 4! ≡ 24 ≡ 6 (mod 9) 5! ≡ 5 · 6 ≡ 30 ≡ 3 (mod 9) Thus, (1! + 2! + 3! + 4! + 5!) ≡ 1 + 2 + 6 + 6 + 3 ≡ 18 ≡ 0 (mod 9) so the remainder is 0 ...
4 List Comprehensions (1) - Homepages | The University of Aberdeen
... • As a larger example, consider the lazy evaluation of the infinite Hamming series, named after Dr. Hamming of Bell Labs: – a series of integers in ascending order; – the first number in the series is 1; – if x is a member of the series, then so is 2 × x, 3 × x and 5 × x ...
... • As a larger example, consider the lazy evaluation of the infinite Hamming series, named after Dr. Hamming of Bell Labs: – a series of integers in ascending order; – the first number in the series is 1; – if x is a member of the series, then so is 2 × x, 3 × x and 5 × x ...
RELATED PROBLEMS 663
... 7-adic integers as coefficients. Consequently neither Bi(n) nor B2(n) may assume the value c more than once. Hence No integer appears in the sequence {a„} more than three times. The preceding discussion does not determine the exact number of times an integer c occurs in the sequence {an}. In order t ...
... 7-adic integers as coefficients. Consequently neither Bi(n) nor B2(n) may assume the value c more than once. Hence No integer appears in the sequence {a„} more than three times. The preceding discussion does not determine the exact number of times an integer c occurs in the sequence {an}. In order t ...
Explicit formulas for Hecke Gauss sums in quadratic
... for the ordinary Gauss sums and ordinary quadratic reciprocity, which, however, follows from (2). It is remarkable, that hence, as a consequence, the quadratic reciprocity law for quadratic number fields is not a genuine new reciprocity law. This is in contrast to what is suggested by Hecke’s proof ...
... for the ordinary Gauss sums and ordinary quadratic reciprocity, which, however, follows from (2). It is remarkable, that hence, as a consequence, the quadratic reciprocity law for quadratic number fields is not a genuine new reciprocity law. This is in contrast to what is suggested by Hecke’s proof ...
Number Theory II: Congruences
... What makes congruences so useful is that, to a large extent, they can be manipulated like ordinary equations. Congruences to the same modulus can be added, multiplied, and taken to a fixed positive integral power; i.e., for any a, b, c, d ∈ Z and m ∈ N we have: • Adding/subtracting congruences: If a ...
... What makes congruences so useful is that, to a large extent, they can be manipulated like ordinary equations. Congruences to the same modulus can be added, multiplied, and taken to a fixed positive integral power; i.e., for any a, b, c, d ∈ Z and m ∈ N we have: • Adding/subtracting congruences: If a ...