A pr 2 00 8 Preprint , arXiv : 0803 . 3737 CONJECTURES ON SUMS OF PRIMES AND TRIANGULAR NUMBERS
نویسندگان
چکیده
In this paper we study integers of 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 integers of the form (2ap− r)/m+ Tx with p a prime. Several conjectures are raised with related analysis and numerical data; our main conjecture asserts that each natural number n 6= 216 can be written in the form p + Tx with p a prime or zero.
منابع مشابه
0803 . 3737 Conjectures on Sums of Primes and Triangular Numbers
We raise the following new conjecture: Each natural number n 6= 216 can be written in the form p + Tx, where p is 0 or a prime, and Tx = x(x + 1)/2 is a triangular number. Some extensions and variants of this conjecture are also discussed. 1. On integers of the form p + Tx Since 1+ 2+ · · ·+ n = n(n+1)/2, those integers Tx = x(x+1)/2 with x ∈ Z are called triangular numbers. Here is a list of t...
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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 (2ap−r)/m+Tx with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with rel...
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