منابع مشابه
On Universal Sums of Polygonal Numbers
For m = 3, 4, . . . , the polygonal numbers of order m are given by pm(n) = (m−2) ` n 2 ́ +n (n = 0, 1, 2, . . . ). For positive integers a, b, c and i, j, k > 3 with max{i, j, k} > 5, we call the triple (api, bpj , cpk) universal if for any n = 0, 1, 2, . . . there are nonnegative integers x, y, z such that n = api(x)+bpj(y)+cpk(z). We show that there are only 95 candidates for universal triple...
متن کاملMixed Sums of Squares and Triangular Numbers
For x ∈ Z let Tx denote the triangular number x(x + 1)/2. Following the recent approach of Z. W. Sun, we show that every natural number can be written in any of the following forms with x, y, z ∈ Z: x + Ty + Tz , x 2 + 2Ty + Tz , x 2 + 3Ty + Tz , x + 5Ty + 2Tz , x 2 + 6Ty + Tz , 3x 2 + 2Ty + Tz , x + 3y + Tz , 2Tx + Ty + Tz , 3Tx + 2Ty + Tz , 5Tx + Ty + Tz . This confirms some conjectures raise...
متن کاملUniversal Mixed Sums of Squares and Triangular Numbers
In 1997 Ken Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form x 2 + y 2 + 10z 2 , equivalently the form 2x 2 + 5y 2 + 4T z represents all integers greater than 1359, where T z denotes the triangular number z(z + 1)/2. Given positive integers a, b, c we...
متن کاملOn Almost Universal Mixed Sums of Squares and Triangular Numbers
In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form x2 + y2+10z2; equivalently the form 2x+5y+4Tz represents all integers greater than 1359, where Tz denotes the triangular number z(z+1)/2. Given positive integers a, b, c we employ modular for...
متن کاملpress . MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS ( III )
In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p = 2m + 1 is a prime congruent to 3 modulo 4 if and only if Tm = m(m + 1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p = x+8(y+z) for no odd integers x, y, z. We also sh...
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ژورنال
عنوان ژورنال: Advances in Pure Mathematics
سال: 2016
ISSN: 2160-0368,2160-0384
DOI: 10.4236/apm.2016.64019