Prime exploration and notebook problems¶
Numerisect 0.5.0 extends this workspace with rigorously proven Mersenne, Fermat, Cullen, Woodall, Wagstaff, decimal-repunit, primorial ± 1, and factorial ± 1 family searches, both Cunningham-chain recurrences, and exact-bit-length NTT-friendly primes. These operations are implemented in PARI/GP and preserve an explicit incomplete status when a result or candidate limit is reached.
Eligible 64-bit interval enumeration now streams from multithreaded primesieve. Larger arbitrary-precision endpoints continue through PARI/GP and every candidate is proven before display. Exact indexed primes and prime counts use parallel primecount when available.
Numerisect imports the useful operations from primes_gap.ipynb and
prime_problems.ipynb without retaining their Python/SymPy calculations or
dependence on a pre-generated prime file. PARI/GP generates and proves primes,
factors integers, and computes statistics. JavaScript only renders the native
results.
Navigation¶
Use Patterns & distribution for gap statistics (/#primes/gap-statistics)
and Goldbach (/#primes/goldbach), Prime generation for primorials
(/#primes/primorial) and random-range sampling (/#primes/random-range),
and Advanced explorations for contiguous digits (/#primes/contiguous-digits)
and bounded problems (/#primes/prime-problem). Each operation occupies one
page with its result and saved-report path immediately below the form.
Prime-gap statistics¶
The existing gap analyzer lists consecutive prime pairs. The new distribution tool additionally calculates, inside PARI/GP:
- exact frequencies for every observed gap size;
- minimum and maximum;
- exact rational mean and median; and
- the mode and its frequency.
The browser draws a frequency chart from that table. A result limit bounds the
number of consecutive gaps scanned per request and returns truncation status.
No external primes_to_1e7_list.txt file is required.
Primorials¶
The primorial generator starts with 2 and repeatedly obtains the next proven
prime, producing 2, 6, 30, 210, 2310, and so on. Products use PARI's
arbitrary-precision integers. Up to 500 cumulative products may be requested.
General prime operations reused¶
The notebooks' prime range/list/generator cells reuse Numerisect's existing
range endpoint. Primes near a number (/#primes/prime-nearby) offers
Find the nth prime and List consecutive primes, in either direction.
It excludes the starting integer and reports exhaustion when a backward search
has too few primes. General k-gap pairs use custom tuple offsets such as [0,40].
For deterministic searches within one modular residue class, including exact sums and continuation points, see Prime manipulation.
The random-range tool fills one missing variant: it samples distinct primes
inside arbitrary-precision endpoints and rigorously rechecks every value with
isprime before returning it.
Digit substrings and Goldbach partitions¶
The contiguous-digit tool tests every distinct substring of the absolute decimal input, without reordering its digits. Inputs are limited to 1,000 digits because the number of substrings is quadratic.
The Goldbach tool accepts one even integer greater than two and returns unique
pairs p ≤ q with p+q=n, rigorously proving both members. This is a bounded
verification for the supplied number and is never presented as a proof of the
Goldbach conjecture.
Four bounded problem searches¶
| Problem | Native condition |
|---|---|
| Prime-square equation | Find primes r<q<p satisfying p²+1=q²+r². |
| Fourth-power sums | Find prime values a⁴+b⁴ with positive a≤b. |
| Three prime factors | Find n for which Ω(n²−1)=3, counting multiplicity. |
| Square divisor sum | Among the first requested prime candidates, find p for which σ(p⁴)=p⁴+p³+p²+p+1 is a square. |
The equation search caps p at 100,000 per request; the fourth-power result
bound is at most 10^12; the factor search spans at most 1,000,000 integers;
and the divisor-sum search considers at most the first 1,000,000 primes. Result
limits protect the browser and report size. These are resource controls for
finite searches, not fixed-precision arithmetic limitations.
API routes¶
POST /api/primes/gap-statistics
POST /api/primes/primorials
POST /api/primes/random-range
POST /api/primes/contiguous-digits
POST /api/primes/goldbach
POST /api/primes/problems
Every successful response includes output_file, and the browser explicitly
shows that the report was saved to output/<filename>.
See Installation and versioning for supported host setup.