prime numbers

Bilingual

Sum of low squares
2 months ago
- For an odd prime p, exactly half of the numbers from 1 to p-1 are quadratic residues (squares modulo p).
- A square modulo p is called 'low' if it is less than p/2, and 'high' if greater than p/2.
- For primes p > 3 with p ≡ 3 mod 4, the signature is defined as the sum of all low squares modulo p.
- The signature uniquely determines the prime p; the prime can be recovered as the largest prime factor of 16s + 1, or as (16s+1)/m where m is the smallest in {3,7,11,15} that yields a prime.
Five Minutes of Prime Time
2 months ago
- The author worked at RSA in 2008, where nerd culture thrived with discussions on prime numbers, combinatorics, and puzzles.
- HR organized monthly games; one challenge was to list all prime numbers between 1 and 1000 in 5 minutes.
- The author's team copied the list from a mobile phone and won the cash prize with all 168 primes correct.
- Another team also copied but omitted some numbers to avoid looking like they cheated, though no rules were broken.
- The team spent the prize on pizza and drinks for everyone; the author remembers this silly game vividly despite other games being more interesting.
Progress on Gilbreath’s conjecture
2 months ago
- Gilbreath's conjecture is simple to explain but challenging to prove, with Paul Erdős speculating it would take 200 years to prove.
- The conjecture is difficult because it lacks obvious mathematical tools, unlike well-studied problems such as differential equations.
- Terence Tao and coauthors published a paper analyzing a random model that mimics Gilbreath's calculation, offering heuristic plausibility but noting rigorous proof remains out of reach.
What does the Riemann zeta function have to do with the distribution of primes?
3 months ago
- Euclid proved there are infinitely many primes around 300 BCE, but later mathematicians sought to quantify how common primes are compared to other infinite sets like even numbers or powers of 2.
- Euler investigated whether the sum of the reciprocals of primes (1/2 + 1/3 + 1/5 + ...) is infinite or finite, linking it to the Riemann zeta function via the Euler product formula, which connects number theory and calculus.
- By taking logarithms and using series expansions, Euler showed that the sum of reciprocal primes diverges to infinity, indicating primes are not too sparse, though the sum grows extremely slowly, like log(log(n)).
- Gauss conjectured that the prime-counting function π(N) is approximately N/log(N) or the logarithmic integral Li(N), based on empirical data, providing a probabilistic estimate for primes.
- Riemann extended Gauss's work by deriving an explicit formula for π(x) involving zeros of the zeta function: π(x) ≈ Li(x) - Σ Li(x^ρ), where ρ are non-trivial zeros, with the main term Li(x) dominating.
- The Riemann hypothesis conjectures that all non-trivial zeros of the zeta function have real part 1/2, which would optimize Gauss's approximation and has profound implications for prime distribution but remains unproven.

prime numbers

Sum of low squares

Five Minutes of Prime Time

Progress on Gilbreath’s conjecture

What does the Riemann zeta function have to do with the distribution of primes?