Hello everyone,
I stumbled upon an intriguing (and quite well-presented) >article< that proposes a new analytical approach to prime numbers using the Lambert W function. I thought it would be a perfect topic for discussion here.
The Core Idea (The Article's Contribution):
The author starts from the reverse of Fermat's Little Theorem. For an odd prime p, we have:
2^(p-1) ≡ 1 (mod p) which implies K = (2^(p-1) - 1) / p
where K is an integer. The novel part is that the author then solves for p explicitly using the Lambert W function W(z), arriving at:
p = -W( -ln(2) / (2K) ) / ln(2)
It's a neat and elegant analytical trick. The author also provides an approximate formula:
p ≈ ln(K)/ln(2) + ln(ln(K))/ln(2) + C
where C is a constant (roughly around 2 for the examples tried).
Here's a table demonstrating the idea (using
As you can see, the neighborhood of the approximation consistently contains primes. The potential computational saving is massive. For
Points for Discussion / Critiques:
The article itself might overhype its findings a bit by framing it as a "formula for primes," but if we look past that, there's a genuinely clever idea here. It's probably not a standalone solution but could it be the basis for a highly efficient hybrid search algorithm? What are your thoughts on the validity of the approach and its practical potential?
Links:
I stumbled upon an intriguing (and quite well-presented) >article< that proposes a new analytical approach to prime numbers using the Lambert W function. I thought it would be a perfect topic for discussion here.
The Core Idea (The Article's Contribution):
The author starts from the reverse of Fermat's Little Theorem. For an odd prime p, we have:
2^(p-1) ≡ 1 (mod p) which implies K = (2^(p-1) - 1) / p
where K is an integer. The novel part is that the author then solves for p explicitly using the Lambert W function W(z), arriving at:
p = -W( -ln(2) / (2K) ) / ln(2)
It's a neat and elegant analytical trick. The author also provides an approximate formula:
p ≈ ln(K)/ln(2) + ln(ln(K))/ln(2) + C
where C is a constant (roughly around 2 for the examples tried).
Here's a table demonstrating the idea (using
C = 2):| 10 | ~8.7 | 4 - 13 | 5, 7, 11, 13 |
| 50 | ~19.2 | 14 - 24 | 17, 19, 23 |
| 100 | ~24.8 | 20 - 29 | 23, 29 |
| 500 | ~40.5 | 35 - 45 | 37, 41, 43 |
[th]
K
[/th][th]Approx. p
[/th][th]Search Range (±5)
[/th][th]Primes in Range
[/th]N = 10^10, instead of a sieve working on ~10^10 elements, you'd only need to perform a primality test on about (number of primes) * (window size) ≈ 455e6 * 20 ≈ 9e9 tests, which is already competitive, not to mention the huge memory savings.Points for Discussion / Critiques:
- Not New in Spirit, But Novel in Approach: Using an approximation for the n-th prime (p_n ~ n log n) is standard. The novelty here is the parameterization via K from Fermat's Little Theorem and the use of the Lambert W.
- The Elephant in the Room - Pseudoprimes: The fundamental issue is that K is also an integer for composite numbers satisfying Fermat's base-2 test (e.g., Carmichael numbers). The sequence of K is "polluted." Any practical algorithm using this must account for this.
- The Practical Hurdles: The value of the constant C isn't universal, and the error in the approximation likely grows with K. How do we dynamically choose the search window size reliably?
- Comparison: How does this approximation's accuracy compare to more standard ones like p_n ≈ n(ln n + ln ln n - 1)?
The article itself might overhype its findings a bit by framing it as a "formula for primes," but if we look past that, there's a genuinely clever idea here. It's probably not a standalone solution but could it be the basis for a highly efficient hybrid search algorithm? What are your thoughts on the validity of the approach and its practical potential?
Links:
- Article: https://fermat-lambert.github.io
