﻿ Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers

# Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers

##### نویسنده
• Yong-Gao Chen
##### چکیده

In this paper, we prove that there is an arithmetic progression of positive odd numbers for each term M of which none of five consecutive odd numbers M,M − 2,M − 4,M − 6 and M − 8 can be expressed in the form 2n ± pα, where p is a prime and n, α are nonnegative integers.

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## M 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...

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## 2 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...

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## press . 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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## 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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## On Silverman's conjecture for a family of elliptic curves

Let \$E\$ be an elliptic curve over \$Bbb{Q}\$ with the given Weierstrass equation \$ y^2=x^3+ax+b\$. If \$D\$ is a squarefree integer, then let \$E^{(D)}\$ denote the \$D\$-quadratic twist of \$E\$ that is given by \$E^{(D)}: y^2=x^3+aD^2x+bD^3\$. Let \$E^{(D)}(Bbb{Q})\$ be the group of \$Bbb{Q}\$-rational points of \$E^{(D)}\$. It is conjectured by J. Silverman that there are infinitely many primes \$p\$ for which \$...

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##### عنوان ژورنال:
• Math. Comput.

دوره 74  شماره

صفحات  -

تاریخ انتشار 2005