Mixed Sums of Squares and Triangular Numbers
نویسندگان
چکیده
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 raised by Sun.
منابع مشابه
Arith . MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS
By means of q-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus x+y for some x, y ∈ Z with x 6≡ y (mod 2) or x = y > 0. The paper also contains some other results and open conjectures on mixed sums of squares and triangular numbers.
متن کاملMixed Sums of Squares and Triangular Numbers 3
By means of q-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus x + y for some integers x > 0 and y > 0 with x 6≡ y (mod 2) or x = y. The paper also contains some other results and open conjectures on mixed sums of squares and triangular numbers.
متن کامل. N T ] 1 0 Ju l 2 00 6 MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS ( II )
For x ∈ Z let T x 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:
متن کامل2 00 8 Preprint ( April 23 , 2008 ) MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS ( III )
In this paper we prove 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 mod 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 sho...
متن کاملMixed 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...
متن کاملM ay 2 00 8 Preprint , arXiv : 0804 . 3750 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...
متن کامل