Perfect Number Test

Six is the first number that sums its own parts: 6 = 1 + 2 + 3. Twenty-eight does the same with more pieces: 28 = 1 + 2 + 4 + 7 + 14. Numbers with that lucky equality are perfect - and they are staggeringly rare. The known ones run 6, 28, 496, 8128, 33550336, 8589869056... every one even, every one of the Euclid-Euler form 2^(p-1) times a Mersenne prime 2^p - 1, and every one exact, because sigma is computed from prime factors, never estimated.

Perfect Number Test factorizes any value up to a trillion by trial division, computes sigma(n) as the product (p^(e+1) - 1)/(p - 1), and reports one of three verdicts - perfect, abundant or deficient - with the full arithmetic: factorization, sigma, proper-divisor sum and, for perfect inputs, the matched Mersenne exponent. Batches of 10,000 run in one pass, all in the browser.

• Exact sigma: Divisor sums from prime factorizations - (p^(e+1) - 1)/(p - 1) per prime, never an estimate.
• Three truthful verdicts: Perfect, abundant or deficient, shown with the comparison that decides.
• Full arithmetic row: Factorization, sigma, proper sum, and excess or deficit on every line.
• Mersenne form detector: Even perfects report their p with 2^(p-1)(2^p - 1).
• Batch testing: Up to 10,000 values per run, each with independent verdict and detail.
• Trillion ceiling: Inputs of up to twelve digits are fully covered by the prime table.
• Copy or download: Verdict list and numbers list, both one-click exports.
• Reference tab: The known perfect numbers, the Euclid-Euler theorem, open odd cases.
• Free and private: Everything runs locally; nothing is uploaded or stored.

1. Enter numbers - one per line, or a batch separated by commas or spaces.
2. Read the verdict badge - perfect, abundant, or deficient.
3. See the sigma arithmetic - sigma, proper-divisor sum and the deciding comparison.
4. Catch the exponent - even perfects display their Mersenne p.
5. Copy or download the results for research, teaching or further analysis.

Example 1 - The first two. Enter 6 and 28. Both return PERFECT: sigma(6) = 12 = 2 x 6 and sigma(28) = 56 = 2 x 28, with proper sums 6 and 28.

Example 2 - The giants. Enter 33550336 and 8589869056. The tool factorizes both instantly and reports PERFECT with the exponents 13 and 17 - 2^12(2^13 - 1) and 2^16(2^17 - 1).

Example 3 - Near misses. Enter 12 and 16. 12 is abundant: sigma is 28, proper sum 16 beats 12. 16 is deficient: sigma is 31, proper sum 15 misses by one.

Example 4 - Nothing is assumed. Enter 945 and 2048. 945 is abundant and odd; 2048 is deficient - sigma 4095, proper 2047, short by one. Both verdicts come from computed factors.

Example 5 - Batch audit. Paste 6, 12, 28, 45, 496, 1000, 8128, 945, 10089 together: the badges mix perfect, abundant and deficient with factorization detail on every row.

• Proven verdicts: Every classification rests on exact sigma arithmetic, not tables.
• Rare-number focus: Full support for every known perfect below a trillion, with p-forms shown.
• Transparent math: Factorization to sigma to comparison - the whole proof on the row.
• Batch speed: Ten thousand candidates classified per run with per-line detail.
• Open-problem aware: Odd inputs get computed answers, with the honest status in the reference.
• Private and free: Everything runs in the browser; nothing is uploaded, stored or logged.