Ramanujan’s theta functions and sums of triangular numbers
نویسندگان
چکیده
منابع مشابه
On Sums of Primes and Triangular Numbers
We study whether sufficiently large integers can be written in the form cp+ Tx, where p is either zero or a prime congruent to r mod d, and Tx = x(x + 1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2p−r)/m+Tx with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with rela...
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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...
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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...
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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...
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2019
ISSN: 1793-0421,1793-7310
DOI: 10.1142/s1793042119500520