5. Solving Linear Congruences
... (1) If linear congruence ax ≡ b (mod n) has a solution, then there are always infinitely many solutions, because if x0 is a solution, then x0 + kn is also a solution for any integer k. By mod n, we can always assume that 0 ≤ x0 ≤ n − 1. In other words, the solution of a linear congruence ax ≡ b (mod ...
... (1) If linear congruence ax ≡ b (mod n) has a solution, then there are always infinitely many solutions, because if x0 is a solution, then x0 + kn is also a solution for any integer k. By mod n, we can always assume that 0 ≤ x0 ≤ n − 1. In other words, the solution of a linear congruence ax ≡ b (mod ...
A Transition to Advanced Mathematics
... These first four chapters contain the core material of the text and, in addition, offer the opportunity for further work in several optional sections: basics of number theory (Section 1.7), combinatorial counting (Section 2.6), order relations and graph theory (Sections 3.4 and 3.5), and image sets ...
... These first four chapters contain the core material of the text and, in addition, offer the opportunity for further work in several optional sections: basics of number theory (Section 1.7), combinatorial counting (Section 2.6), order relations and graph theory (Sections 3.4 and 3.5), and image sets ...
Here
... How many different four-letter words, including nonsense words, can be produced by rearranging the letters in LUCK? In the absence of a more inspired approach, there is always the brute-force strategy: Make a systematic list. Once we become convinced that Fig. 1.1.1 accounts for every possible rearr ...
... How many different four-letter words, including nonsense words, can be produced by rearranging the letters in LUCK? In the absence of a more inspired approach, there is always the brute-force strategy: Make a systematic list. Once we become convinced that Fig. 1.1.1 accounts for every possible rearr ...
Elementary Number Theory - science.uu.nl project csg
... Most of us have heard about them at a very early age. We also learnt that there are infinitely many of them and that every integer can be written in a unique way as a product of primes. These are properties that are not mentioned in our rules. So one has to prove them, which turns out to be not entir ...
... Most of us have heard about them at a very early age. We also learnt that there are infinitely many of them and that every integer can be written in a unique way as a product of primes. These are properties that are not mentioned in our rules. So one has to prove them, which turns out to be not entir ...
The largest prime factor of a Mersenne number
... Theorem 2. (GRH) But for o(π(x)) primes p ≤ x we have P (2p − 1) > p4/3 /(log p)2/3 log log p. Let Φn (x) denote the nth cyclotomic polynomial. Theorem 3. (GRH) Uniformly for each > 0 and x ≥ 2, but for O(x + x/(log x) 2 ) integers n ≤ x, every prime factor of Φn (2) exceeds n1+ . Further, but f ...
... Theorem 2. (GRH) But for o(π(x)) primes p ≤ x we have P (2p − 1) > p4/3 /(log p)2/3 log log p. Let Φn (x) denote the nth cyclotomic polynomial. Theorem 3. (GRH) Uniformly for each > 0 and x ≥ 2, but for O(x + x/(log x) 2 ) integers n ≤ x, every prime factor of Φn (2) exceeds n1+ . Further, but f ...
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"".