In a recent post, I proposed a conjecture suggesting that every non-zero integer can be expressed as the difference between a semiprime and a prime.
Conjecture: Every Non-Zero Integer as the Difference of a Semiprime and a Prime
To explore this idea further, I examined a related but smaller problem. This exploration led to the following new conjecture, now including zero and focusing on non-negative integers:
(Note: The range of the conjecture was incorrect, so the title and content have been revised.)
Conjecture: For any prime number $p$, every non-negative integer $k$ (including zero) can be expressed in the form: $$ k = p \cdot m - q $$
where $m$ is a positive integer and $q$ is a prime number (which may be the same as or different from $p$).
Computational Evidence
To investigate this conjecture, I conducted a large-scale computational experiment using GPU acceleration. The steps were as follows:
1.Prime Generation: Generated all primes up to a very large limit ($p_{\text{max}}$) and used this set for the multiples $p$. Created a prime bitmask for rapid prime checks up to $q_{\text{limit}}=101,010,000$.
2.Expressibility Check: For each integer $k$ in the range $[0,100,000,000]$ (including zero), checked whether there exists a prime $p$ (from 2 up to $p_{\text{max}}$) and an integer $m≥1$ such that $k=p⋅m−q$ for some prime $q$. Utilized GPU parallel processing to handle the extensive computations.
Results: Prime Generation:
=== Result ===
All integers in the range [0, 100,000,000], including zero, can be expressed as p * m - q , where p is any prime between 2 and 383,393, m is a positive integer, and q is any prime.
This computational evidence suggests that the conjecture holds for integers from 0 to $100,000,000$ and for any prime $p$ from $2$ to $383,393$. Although this does not constitute a formal proof, it provides a strong numerical basis for considering the conjecture's validity in general.
Questions and Discussion Points
Theoretical Implications:
What mathematical properties of primes might underlie the ability to express every non-negative integer, including zero, in this form?
Could this conjecture be connected to or derived from existing results in number theory, such as prime gaps or properties of prime distributions?
Relationship to Semiprimes:
While this conjecture does not directly involve semiprimes, understanding why any prime p allows for such an expression could provide insights into the original conjecture involving semiprimes.
Formal Proof or Counterexamples:
What approaches might one take to prove or disprove this conjecture for all primes p?
Are there known counterexamples or classes of numbers that are not expressible in this form?
Further Computational Work:
Should further computational experiments extend beyond the current range or explore additional properties of p and q to gather more evidence?
Conclusion
This numerical investigation, including the ability to express zero and all positive integers in the specified form, hints at a potentially deep relationship between integers, prime multiples, and prime differences. While the computational evidence is strong, a formal proof or theoretical framework is necessary to fully understand this phenomenon. I am curious to hear thoughts from the community on possible approaches to explore or prove this conjecture.
