On triangular numbers which are sums of consecutive squares
نویسندگان
چکیده
منابع مشابه
Mixed 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...
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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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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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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 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: Journal of Number Theory
سال: 1972
ISSN: 0022-314X
DOI: 10.1016/0022-314x(72)90036-4