Prime quadruplet finder

Enter an integer of at least thirteen and receive an ordered collection of quadruplets together with a count. The calculation uses a deterministic sieve, so repeated requests with the same limit produce exactly the same result. It is useful for exploring prime constellations, checking examples, preparing exercises, and generating dependable test data without manually testing each candidate.

What the finder includes in its results

A prime quadruplet in this tool has one precise shape: four prime numbers written as p, p plus two, p plus six, and p plus eight. The limit n is inclusive and applies to the complete pattern. In other words, a quadruplet is returned only when its largest member, p plus eight, is less than or equal to n. This interpretation makes boundary behavior predictable: a pattern does not appear early merely because its first prime is within the limit. The first valid result is 5, 7, 11, 13, which is why the smallest accepted input is thirteen. Output is sorted by the first prime because candidates are examined in ascending order. Each result is represented as a four-number array, and the count field tells you how many arrays were found. The tool does not list arbitrary groups of four primes, near matches, or arrangements with different gaps; every returned row follows the exact offsets zero, two, six, and eight.

How the calculation works

The calculator first validates that n is a whole number within the published range. It then builds a Sieve of Eratosthenes from zero through n. The sieve marks composite numbers by starting with each unmarked prime and crossing out its multiples, beginning at the square of that prime. After this bounded pass, primality checks are direct lookups rather than repeated trial divisions. The algorithm scans possible starting values p only while p plus eight remains inside the requested limit. For each candidate it checks the four required positions and adds the pattern only when all four are prime. This approach is deterministic: it uses no network service, randomness, current time, stored state, or probabilistic primality test. Consequently, the same valid input always gives the same ordered JSON output. The fixed maximum protects the browser and API worker from unbounded memory and execution time while still allowing substantial lists for experimentation, demonstrations, and automated verification.

Using and interpreting the output

Choose n according to the largest number you want included, not merely the largest starting prime you want tested. For example, if a candidate begins at p, it can appear only when n reaches p plus eight. Read the quadruplets field as an ascending list of independent patterns; each nested array contains exactly four integers. The count is the length of that list and is convenient when a script needs a summary without recounting rows. A valid search may return an empty list when no complete pattern lies inside its range, although the minimum accepted boundary already contains the first known pattern. Use the result directly in lessons about prime gaps, in regression fixtures for mathematical software, or as input to a separate statistical analysis. Remember that the finder enumerates a specific constellation and does not prove broader conjectures about whether infinitely many such patterns exist. API automation costs $0.002 per request, while the same pure calculation can run through the page’s browser experience.