CANT 2015, CANT 2016: Combinatorial and Additive Number Theory II pp 279-310

Conjectures on Representations Involving Primes

• Zhi-Wei Sun
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 220)

Abstract

We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer n, there exists $$k\in \{0,\ldots ,n\}$$ such that $$n+k$$ and $$n+k^2$$ are both prime. (ii) Each integer $$n>1$$ can be written as $$x+y$$ with $$x,y\in \{1,2,3,\ldots \}$$ such that $$x+ny$$ and $$x^2+ny^2$$ are both prime. (iii) For any rational number $$r>0$$, there are distinct primes $$q_1,\ldots ,q_k$$ with $$r=\sum _{j=1}^k1/(q_j-1)$$. (iv) Every $$n=4,5,\ldots$$ can be written as $$p+q$$, where p is a prime with $$p-1$$ and $$p+1$$ both practical, and q is either prime or practical. (v) Any positive rational number can be written as m/n, where m and n are positive integers with $$p_m+p_n$$ a square (or $$\pi (m)\pi (n)$$ a positive square), $$p_k$$ is the kth prime and $$\pi (x)$$ is the prime-counting function.

Keywords

Conjectures Primes Practical numbers Representations

2010 Mathematics Subject Classification.

Primary 11A41 11P32 Secondary 11B13 11D68

Notes

Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant No. 11571162).

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