Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series

In this article we shall examine several different types of figurative numbers which have been studied extensively over the period of 2500 years, and currently scattered on hundreds of websites. We shall discuss their computation through simple recurrence relations, patterns and properties, and mutual relationships which have led to curious results in the field of elementary number theory. Further, for each type of figurative numbers we shall show that the addition of first finite numbers and infinite addition of their inverses often require new/strange techniques. We sincerely hope that besides experts, students and teachers of mathematics will also be benefited with this article.

Two Inequalities for Convex Functions

Let a_0 < a_1 < … < a_n be positive integers with sums Σ_(i=0)~n∈_ia_i(∈_i = 0,1) distinct. P. Erdos conjectured that Σ_(i=0)~n 1/a_i ≤ Σ_(i=0)~n 1/2~i. Thebest known result along this line is that of Chen: Let f be any given convex decreasing function on[A, B] with α_0, α_1, …, α_n , β_0, β_1, …, β_n being real numbers in [A, B] with α_0 ≤α_1 ≤ … ≤ α_n, Σ_(i=0)~n α_i ≥ Σ_(i=0)~n β_i, k = 0, …, n. Then Σ_(i=0)~n f(α_i) ≤Σ_(i=0)~n f(β_i). In this paper, we obtain two generalizations of the above result; each is ofspecial interest in itself. We prove:Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α_0,α_1, …, α_n , β_0, β_1, …, β_n, α'_0, α'_1, …, α'_n , β'_0, β'_1, …, β'_n be realnumbers in [A, B] with α'_0 ≤ α'_1 ≤ … ≤ α_n. Then Σ_(i=0)~n f(α_i)g(α'_i) ≤ Σ_(i=0)~nf(β_i)g(β'_i), k = 0, …, n. Theorem 2 Let f be any given convex decreasing function on [A, B]with k_0, k_1, …, k_n being nonnegative real numbers and α_0, α_1, …, α_n , β_0, β_1, …,β_n being real numbers in [A, B] with α_0 ≤ α_1 ≤ … ≤ α_n, Σ_(i=0)~t k_i α_i ≥ Σ_(i=0)~tk_iβ_i, t = 0, …, n. Then Σ_(i=0)~t k_if(α_i) ≤ Σ_(i=0)~t k_if_(β_i).