Perfect Numbers Generator

Among the most ancient objects in number theory: integers that are exactly the sum of their own proper divisors. Six, twenty-eight, four-hundred-ninety-six — each one is a closed arithmetic circle, and beyond 8128 the numbers explode: 33,550,336, 8,589,869,056, 137,438,691,328...

Perfect Numbers Generator produces them by the deciding theorem: every even perfect number is 2^(k-1)(2^k - 1) with k prime and 2^k - 1 prime (Mersenne primes). Each exponent and each Mersenne factor is checked in-browser with exact BigInt Miller-Rabin, so no list is pre-fabricated — the numbers are discovered, k by k, while you watch.

• Discovery, not lookup: Mersenne primes are found by live Miller-Rabin primality tests over native BigInt arithmetic.
• Exact huge values: The 12th perfect number, all 77 digits, computed and rendered precisely.
• Evidence rows: Each result shows its exponent k and the Mersenne factor 2^k - 1 together with its primality verdict.
• Digit counts: Values annotated with length so the exponential growth is visible at a glance.
• Reference tab: Euclid-Euler proof sketch, divisor statistics, the odd-perfect question.
• Free and private: All testing is local; nothing is uploaded, stored or logged.

1. Choose the count — how many consecutive perfect numbers to show (1 to 12).
2. Generate — the tool scans prime k from 2 upward, testing each Mersenne number 2^k - 1.
3. Read the rows — exponent, Mersenne prime, exact value and digit count.
4. Marvel at growth — the digits roughly double after the seventh term.
5. Copy or download — the full list for analysis.

Example 1 — The first four. Count 4: 6, 28, 496, 8128. Divisors: 1+2+3=6; 1+2+4+7+14=28; the pattern continues exactly.

Example 2 — A big gap. Generating 6 terms jumps from 8128 to 33,550,336: exponents 7, 13, 17 — every intermediate prime k fails because 2^k - 1 is composite.

Example 3 — The familiar giant. Count 8 ends with 2,305,843,009,213,693,952 (k = 31), a value the Greeks never reached; BigInt renders it exactly.

Example 4 — Over a quintillion digits. Term 10 (k = 89) has 54 digits — 19 quintillion more digits than term 5, and still exact.

Example 5 — The 12th. Generate all 12: the last, k = 127, spans 77 digits, always the same value as Euclid's route via 2^126 (2^127 - 1).

• Mathematically exact: No canned lists; every term is recomputed from prime tests.
• True BigInt: 77-digit values without a single rounding error.
• Transparent: Exponent and Mersenne factor shown for each number.
• Flexible output: Any prefix of the known even perfect list, up to all 12 displayed.
• Educational reference: The theorem, the history, and the open odd-perfect question in one tab.
• Private by design: No server contact; nothing uploaded, stored or logged.