Solving a linear equation in a set of integers I
... Some of these equations have “trivial” solutions that need to be excluded from the consideration; for arithmetical progressions, they were collections of three identical numbers, in Sidon’s problem, collections where (u, v) is a permutation of (x, y). In general, we define trivial solutions as follo ...
... Some of these equations have “trivial” solutions that need to be excluded from the consideration; for arithmetical progressions, they were collections of three identical numbers, in Sidon’s problem, collections where (u, v) is a permutation of (x, y). In general, we define trivial solutions as follo ...
Circular sets of prime numbers and p
... relation structure have cohomological dimension 2. By a result of H. Koch [Ko], GS (Q)(p) has such a relation structure if the set of prime numbers S satisfies a certain technical condition. In this way, Labute obtained first examples of pairs (p, S) with p ∈ / S and cd GS (Q)(p) = 2, e.g. p = 3, S ...
... relation structure have cohomological dimension 2. By a result of H. Koch [Ko], GS (Q)(p) has such a relation structure if the set of prime numbers S satisfies a certain technical condition. In this way, Labute obtained first examples of pairs (p, S) with p ∈ / S and cd GS (Q)(p) = 2, e.g. p = 3, S ...
Around the Littlewood conjecture in Diophantine approximation
... In particular, the Littlewood conjecture holds for the pair (α, β) for any β in Bϕ (α). Furthermore, the set Bϕ (α) can be effectively constructed. The proof of Theorem 5 rests on the theory of continued fractions. For given α and ϕ, we construct inductively the sequence of partial quotients of a su ...
... In particular, the Littlewood conjecture holds for the pair (α, β) for any β in Bϕ (α). Furthermore, the set Bϕ (α) can be effectively constructed. The proof of Theorem 5 rests on the theory of continued fractions. For given α and ϕ, we construct inductively the sequence of partial quotients of a su ...
Ramanujan, taxicabs, birthdates, zipcodes and twists
... obtained no bound beyond which every odd integer is so represented. Although obtaining such a bound appears to be beyond the current state of knowledge, assuming certain Riemann hypotheses, the author and Soundararajan [5] have shown that the only positive odd integers that are not of the form x2 + ...
... obtained no bound beyond which every odd integer is so represented. Although obtaining such a bound appears to be beyond the current state of knowledge, assuming certain Riemann hypotheses, the author and Soundararajan [5] have shown that the only positive odd integers that are not of the form x2 + ...
The Pentagonal Number Theorem and All That
... Jordan Bell wrote an interesting paper on the history of the Pentagonal Number Theorem. It can be found at http://arxiv.org/pdf/math/0510054v2. The first mention of the theorem is in a letter from Daniel Bernoulli to Euler on January 28, 1741. Bernoulli is replying to a (lost) letter from Euler abou ...
... Jordan Bell wrote an interesting paper on the history of the Pentagonal Number Theorem. It can be found at http://arxiv.org/pdf/math/0510054v2. The first mention of the theorem is in a letter from Daniel Bernoulli to Euler on January 28, 1741. Bernoulli is replying to a (lost) letter from Euler abou ...
Exercise 1
... Euclid’s algorithm and Bezout’s Lemma. Thus, when proving claims about Euclid’s algorithm (e.g. in the exercises above) we cannot use the fundamental theorem of arithmetic. 5.5.2. Remark: Although finding the greatest common divisor using factorization may seems to be easier than Euclid’s algorithm, ...
... Euclid’s algorithm and Bezout’s Lemma. Thus, when proving claims about Euclid’s algorithm (e.g. in the exercises above) we cannot use the fundamental theorem of arithmetic. 5.5.2. Remark: Although finding the greatest common divisor using factorization may seems to be easier than Euclid’s algorithm, ...
![arXiv:math/0511682v1 [math.NT] 28 Nov 2005](http://s1.studyres.com/store/data/014696627_1-8def914a5ac3ed74bde3727e1309931c-300x300.png)
A clasification of known root prime-generating polynomials
... 16. Some Carmichael numbers are also Harshad numbers but the most of them aren’t. The number 561 has yet another interesting related property; if we note with s(n) the iterated sum of the digits of a number n that not goes until the digital root but stops to the last odd prime obtained before this, ...
... 16. Some Carmichael numbers are also Harshad numbers but the most of them aren’t. The number 561 has yet another interesting related property; if we note with s(n) the iterated sum of the digits of a number n that not goes until the digital root but stops to the last odd prime obtained before this, ...
Revisiting a Number-Theoretic Puzzle: The Census
... implies that b = c = e = f . In both cases, we conclude that {a, b, c} = {d, e, f }, a contradiction. q.e.d. With Theorem 1, we further assume that c > f . More than this assumption, Theorem 1 eliminates a good number positive integers as candidates for CTNs. A prime cannot be a CTN because all trip ...
... implies that b = c = e = f . In both cases, we conclude that {a, b, c} = {d, e, f }, a contradiction. q.e.d. With Theorem 1, we further assume that c > f . More than this assumption, Theorem 1 eliminates a good number positive integers as candidates for CTNs. A prime cannot be a CTN because all trip ...
![arXiv:math/0008222v1 [math.CO] 30 Aug 2000](http://s1.studyres.com/store/data/015263483_1-3fbf9f2fcdc71976e9b3a5d7ef45a284-300x300.png)
arXiv:math/0008222v1 [math.CO] 30 Aug 2000
... Although it is by no means obvious at first glance, this number is always a perfect square or twice a perfect square (see [L]). Furthermore, it is divisible by 2n but no higher power of 2. This fact about 2-divisibility was independently proved by several people (see [JSZ], or see [P] for a combinat ...
... Although it is by no means obvious at first glance, this number is always a perfect square or twice a perfect square (see [L]). Furthermore, it is divisible by 2n but no higher power of 2. This fact about 2-divisibility was independently proved by several people (see [JSZ], or see [P] for a combinat ...
Answers
... (d) True - a square is a special type of rectangle. However, the converse is false, not every rectangle is a square. (e) False (the circumference of a circle is only approximately 3 times the diameter) (f) True ...
... (d) True - a square is a special type of rectangle. However, the converse is false, not every rectangle is a square. (e) False (the circumference of a circle is only approximately 3 times the diameter) (f) True ...
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"".