On integers of the forms k ± 2n and k2 n ± 1
... odd integers. On the other hand, Sierpiński [34] proved that there are infinitely many positive odd numbers k for which all k2n + 1 (n = 1, 2, . . .) are composite. In 1962, J.L. Selfridge (unpublished) discovered that for any positive integer n, the integer 78 557 · 2n + 1 is divisible by one of t ...
... odd integers. On the other hand, Sierpiński [34] proved that there are infinitely many positive odd numbers k for which all k2n + 1 (n = 1, 2, . . .) are composite. In 1962, J.L. Selfridge (unpublished) discovered that for any positive integer n, the integer 78 557 · 2n + 1 is divisible by one of t ...
LecWeek9
... bn=2n-2+bn-2+bn-1 bn=2n-3+bn-3+bn-2+bn-1 . bn= bn-1+bn-2-bn-3 Hn=2Hn-1+1 In each case, many sequences satisfy the relationship and one also needs to set/have initial conditions get a unique solution. Goal: “closed form” expression for function, f, for the sequence, giving the nth term in a formula t ...
... bn=2n-2+bn-2+bn-1 bn=2n-3+bn-3+bn-2+bn-1 . bn= bn-1+bn-2-bn-3 Hn=2Hn-1+1 In each case, many sequences satisfy the relationship and one also needs to set/have initial conditions get a unique solution. Goal: “closed form” expression for function, f, for the sequence, giving the nth term in a formula t ...
Full text
... m — Yli=i ai^i, where a* E S — {0,1}. He also studied the case of the set S = {0,1, • • • , r } in [3]. In this paper we shall study the r-subcomplete partitions which are complete partitions with the set S = {— r, • • • , — 1,0,1, • • • , r } , where r is a positive integer. 2. T H E r - S U B C O ...
... m — Yli=i ai^i, where a* E S — {0,1}. He also studied the case of the set S = {0,1, • • • , r } in [3]. In this paper we shall study the r-subcomplete partitions which are complete partitions with the set S = {— r, • • • , — 1,0,1, • • • , r } , where r is a positive integer. 2. T H E r - S U B C O ...
Cauchy sequences. Definition: A sequence (xn) is said to be a
... constant sequences (r) with rational numbers r) Tn = {k ∈ Z : k/2n is an upper bound of S}. First, let us show that it is nonempty. Let X be an upper bound of S. We may assume that X > 0. Now, note that 1/X > 0 and there exists rational number r = m/K, m, K ∈ N such that 0 < m/K < 1/X. Clearly 1/X > ...
... constant sequences (r) with rational numbers r) Tn = {k ∈ Z : k/2n is an upper bound of S}. First, let us show that it is nonempty. Let X be an upper bound of S. We may assume that X > 0. Now, note that 1/X > 0 and there exists rational number r = m/K, m, K ∈ N such that 0 < m/K < 1/X. Clearly 1/X > ...
Elementary mathematics
Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. The most basic topics in elementary mathematics are arithmetic and geometry. Beginning in the last decades of the 20th century, there has been an increased emphasis on problem solving. Elementary mathematics is used in everyday life in such activities as making change, cooking, buying and selling stock, and gambling. It is also an essential first step on the path to understanding science.In secondary school, the main topics in elementary mathematics are algebra and trigonometry. Calculus, even though it is often taught to advanced secondary school students, is usually considered college level mathematics.