abstract - Universiteit Leiden
... Baker [2, 1968] gave in the case A = Z an effective proof that (1) has only finitely many solutions. This was extended by Coates [7, 1968/69] to the case A = Z[(p1 · · · pt )−1 ] where the pi are distinct primes and by Kotov and Sprindzhuk [10, 1973] to the case that A is the ring of S-integers in a ...
... Baker [2, 1968] gave in the case A = Z an effective proof that (1) has only finitely many solutions. This was extended by Coates [7, 1968/69] to the case A = Z[(p1 · · · pt )−1 ] where the pi are distinct primes and by Kotov and Sprindzhuk [10, 1973] to the case that A is the ring of S-integers in a ...
2016 UI UNDERGRADUATE MATH CONTEST Solutions
... Solution. Let N be the given integer, and consider the integers Nk , k = 0, 1, . . . , 2015, obtained from N by replacing the last k digits by 0’s. (In particular, N0 = N is the given integer.) Since N has 2016 digits, none of which is 0, the integers Nk are all distinct, non-zero descendants of N0 ...
... Solution. Let N be the given integer, and consider the integers Nk , k = 0, 1, . . . , 2015, obtained from N by replacing the last k digits by 0’s. (In particular, N0 = N is the given integer.) Since N has 2016 digits, none of which is 0, the integers Nk are all distinct, non-zero descendants of N0 ...
4-3: Alternating Series, and the Alternating Series Theorem
... down to a value as n → ∞ (if you’re curious, the above series converges to ln 2 which we’ll prove later in the semester). • Definition: A series is called Pan alternating series if the terms alternate in sign. That is, an alternating series is a series of the form (−1)k+1 ak where ak > 0 for all k. ...
... down to a value as n → ∞ (if you’re curious, the above series converges to ln 2 which we’ll prove later in the semester). • Definition: A series is called Pan alternating series if the terms alternate in sign. That is, an alternating series is a series of the form (−1)k+1 ak where ak > 0 for all k. ...
UNIQUE FACTORIZATION IN MULTIPLICATIVE SYSTEMS
... Pu P2, " • • 1 Ph- Thus the set ilf can be characterized as all positive integers relatively prime to ph+i, • • • , Pr, if such primes exist. So the set ilf can be described in terms of the modulus ph+iph+2 • ■ • pr, which is less than n since h^l. This contradicts our basic hypothesis that n is the ...
... Pu P2, " • • 1 Ph- Thus the set ilf can be characterized as all positive integers relatively prime to ph+i, • • • , Pr, if such primes exist. So the set ilf can be described in terms of the modulus ph+iph+2 • ■ • pr, which is less than n since h^l. This contradicts our basic hypothesis that n is the ...
A Refinement of the Function $ g (m) $ on Grimm Conjecture
... proof, see also [4, Appendix 1]. Furthermore, by Theorem 1 in [1], it is not difficult to prove that Grimm’s Conjecture implies that there are primes between x2 and for x2 +x all sufficiently large x and there are primes between y 2 − y and y 2 for all sufficiently large y. Let x = n and y = n + 1, ...
... proof, see also [4, Appendix 1]. Furthermore, by Theorem 1 in [1], it is not difficult to prove that Grimm’s Conjecture implies that there are primes between x2 and for x2 +x all sufficiently large x and there are primes between y 2 − y and y 2 for all sufficiently large y. Let x = n and y = n + 1, ...
Continued fractions Yann BUGEAUD Let x0,x1,... be real numbers
... a continued fraction, provided that the limit exists. The aim of this Section is to show how a real number ξ can be expressed as ξ = [x0 ; x1 , x2 , . . .], where x0 is an integer and xn a positive integer for any n ≥ 1. We first deal with the case of a rational number ξ, then we describe an algorit ...
... a continued fraction, provided that the limit exists. The aim of this Section is to show how a real number ξ can be expressed as ξ = [x0 ; x1 , x2 , . . .], where x0 is an integer and xn a positive integer for any n ≥ 1. We first deal with the case of a rational number ξ, then we describe an algorit ...
Equivalents of the (Weak) Fan Theorem
... (β)i. Then for all x there exist i,n such that the nth approximation of x is between the nth approximation of (α)i and the nth approximation of (β)i: (α)i(n)00 < x(n)0 ≤ x(n)00 < (β)i(n)0 Then (by making n bigger): for all x there exist i,n such that i < n and the nth approximation of x is between t ...
... (β)i. Then for all x there exist i,n such that the nth approximation of x is between the nth approximation of (α)i and the nth approximation of (β)i: (α)i(n)00 < x(n)0 ≤ x(n)00 < (β)i(n)0 Then (by making n bigger): for all x there exist i,n such that i < n and the nth approximation of x is between t ...
Small solutions to systems of polynomial equations with integer
... mentions availability of Mathematica, MuPAD, Perl and Python codes for checking this by solving randomly chosen systems. However, there is evidence that the statements do not hold for rings relevant in number theory, except maybe for very small n. Matiyasevich’s solution to Hilbert’s tenth problem i ...
... mentions availability of Mathematica, MuPAD, Perl and Python codes for checking this by solving randomly chosen systems. However, there is evidence that the statements do not hold for rings relevant in number theory, except maybe for very small n. Matiyasevich’s solution to Hilbert’s tenth problem i ...
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"".