✨ 完全数生成器 / 判断器Perfect Number Generator & Checker
由已知的梅森素数 2ᵖ − 1 按欧几里得–欧拉公式 2ᵖ⁻¹(2ᵖ − 1) 精确生成前 1–25 个偶完全数（最大 13066 位），或判断一个数是完全数、盈数还是亏数并列出它的真因数。Generate the first 1–25 even perfect numbers exactly from the known Mersenne primes 2ᵖ − 1 with the Euclid–Euler formula 2ᵖ⁻¹(2ᵖ − 1) — the largest has 13,066 digits — or check whether a number is perfect, abundant or deficient and list its proper divisors.
使用说明与常见问题How-to & FAQ
「完全数生成器 / 判断器」是什么？What is the Perfect Number Generator & Checker tool?
由已知的梅森素数 2ᵖ − 1 按欧几里得–欧拉公式 2ᵖ⁻¹(2ᵖ − 1) 精确生成前 1–25 个偶完全数（最大 13066 位），或判断一个数是完全数、盈数还是亏数并列出它的真因数。Generate the first 1–25 even perfect numbers exactly from the known Mersenne primes 2ᵖ − 1 with the Euclid–Euler formula 2ᵖ⁻¹(2ᵖ − 1) — the largest has 13,066 digits — or check whether a number is perfect, abundant or deficient and list its proper divisors.
如何使用「完全数生成器 / 判断器」？How do I use the Perfect Number Generator & Checker tool?
在「完全数生成器 / 判断器」中，填写「模式」、「个数 k（1–25）」、「要判断的数（1 – 10¹²）」等参数，结果会随输入实时计算并显示在下方。In the Perfect Number Generator & Checker: Fill in "Mode", "How many, k (1–25)", "Number to check (1 – 10¹²)"; results are recalculated instantly as you type.
完全数是什么？What is a perfect number?
完全数是等于它所有真因数（除自身以外的正因数）之和的正整数，比如 6 = 1 + 2 + 3，28 = 1 + 2 + 4 + 7 + 14。真因数之和大于它本身的叫盈数（如 12），小于的叫亏数（如 8，以及所有素数）。A perfect number equals the sum of its proper divisors (the positive divisors other than itself): 6 = 1 + 2 + 3, 28 = 1 + 2 + 4 + 7 + 14. If the divisors add up to more it is abundant (12), and if less it is deficient (8, and every prime).
为什么说“偶完全数”？有奇完全数吗？Why 'even' perfect numbers? Are there odd ones?
欧几里得–欧拉定理证明：偶完全数恰好就是 2ᵖ⁻¹(2ᵖ − 1)，其中 2ᵖ − 1 是梅森素数。至今没有人找到奇完全数，也没有人证明它不存在；已知如果存在，它会大于 10¹⁵⁰⁰（Ochem 与 Rao，2012）。所以列表是“前 k 个偶完全数”。The Euclid–Euler theorem shows the even perfect numbers are exactly 2ᵖ⁻¹(2ᵖ − 1) with 2ᵖ − 1 a Mersenne prime. No odd perfect number has ever been found, and none has been ruled out; any that exists is known to exceed 10¹⁵⁰⁰ (Ochem and Rao, 2012). That is why the list says 'the first k even perfect numbers'.
梅森素数指数从哪里来？Where do the Mersenne exponents come from?
前 25 个梅森素数指数 p = 2, 3, 5, 7, 13, … , 19937, 21701 取自 OEIS A000043 和 GIMPS 的已知梅森素数列表，这一段已被完整验证、中间没有遗漏。完全数本身在浏览器里用大整数精确计算。The first 25 Mersenne exponents p = 2, 3, 5, 7, 13, …, 19937, 21701 come from OEIS A000043 and the GIMPS list of known Mersenne primes; this range has been fully verified with nothing missing in between. The perfect numbers themselves are computed exactly in your browser with big integers.
判断模式最大能算多大的数？How large a number can the checker handle?
最大 10¹²。工具从 1 试除到 √n 找出全部因数，10¹² 只需试一百万次，瞬间完成；更大的数会超出 JavaScript 能精确表示整数的范围或耗时太久。Up to 10¹². It finds every divisor by trial division up to √n, which is only a million steps at 10¹² and finishes instantly; larger numbers would exceed the range where JavaScript represents whole numbers exactly, or take too long.
计算精度如何？How precise are the results?
「完全数生成器 / 判断器」使用 JavaScript 双精度浮点数（约 15–16 位有效数字），部分功能使用 BigInt 处理大整数。The Perfect Number Generator & Checker uses double-precision floating point (about 15–16 significant digits); some features use BigInt for large integers.
「完全数生成器 / 判断器」免费吗？需要注册吗？Is the Perfect Number Generator & Checker free? Do I need an account?
「完全数生成器 / 判断器」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Perfect Number Generator & Checker is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.