✖️ 把整数改写成若干数之积Rewrite Integer as Product

使用说明与常见问题How-to & FAQ

「把整数改写成若干数之积」是什么？What is the Rewrite Integer as Product tool?

把每个大于 1 的整数改写成因数乘积：质因数分解，或拆成若干大于 1 的因数（随机，可设种子）。本地处理。Rewrite each integer greater than 1 as a product of factors: prime factorization, or a random factor split (optional seed). Processed locally.

如何使用「把整数改写成若干数之积」？How do I use the Rewrite Integer as Product tool?

在「把整数改写成若干数之积」中，把内容粘贴或输入到上方输入框，按需设置「方式」、「种子（随机）」，结果会实时显示在下方，点击「复制结果」即可复制。例如输入「12」，得到「12 = 2 x 2 x 3」。Paste integers greater than 1 and choose a "Mode": Prime factors gives the prime factorisation (12 = 2 x 2 x 3) and Random factors splits it into random factors; a "Seed" makes the random split repeatable. Click "Copy result" to copy the output.

怎样输入多个整数？How do I enter several integers?

每行一个，或用空格、逗号、分号分隔。像 1,234,567 这样每 3 位一个逗号的写法按千位分隔读作一个数（所以 10,100 是 10100，要表示两个数请写 10 100）；也可以用下划线分组（1_000）。负号可用 -、−（U+2212）或全角 －，全角数字也能识别。小数、科学计数法（1e3）或夹杂文字会报错并指出是第几个、完整显示那个值。One per line, or separated by spaces, commas or semicolons. A value with a comma every three digits, like 1,234,567, is read as one number with thousands separators (so 10,100 is 10100; write 10 100 for two values); underscores also group digits (1_000). Minus can be a hyphen, the Unicode minus (U+2212) or the fullwidth minus, and fullwidth digits work. Decimals, scientific notation (1e3) or stray text give an error naming the position and showing the whole value.

结果精确吗？Are the results exact?

精确。所有整数都按大整数（BigInt）逐位处理，不经过浮点，超过 2^53 的数（例如 9007199254740993）也不会被舍入。粘贴几千上万行也是线性处理，通常在几十毫秒内完成。Yes. Every value is handled as a big integer (BigInt) with no floating-point step, so numbers above 2^53 such as 9007199254740993 are never rounded. Pasting thousands of lines is processed in linear time, usually within tens of milliseconds.

怎样分解大数？How are large numbers factored?

先用小质数试除，再判断是否为质数（前 13 个质数为底的 Miller–Rabin，在 3.3×10^24 以下是确定性的；更大的数再加强 Lucas 检验，即 Baillie–PSW，目前没有已知反例，这样的因数标注「BPSW 概率质数」），合数用 Pollard rho 算法找因数，全程大整数运算。2^61−1 这样的大质数或两个十位质数的乘积通常不到 1 秒。如果某个因数太大、在时限内找不出，会原样保留并标注「未能分解」，不会卡住页面。Small primes are tried first, then primality is tested (Miller–Rabin with the first 13 prime bases, exact below 3.3×10^24; larger values also get a strong Lucas test, i.e. Baillie–PSW, which has no known counterexample, and such factors are marked BPSW probable prime) and Pollard's rho finds factors of composites, all in big-integer arithmetic. A large prime such as 2^61−1, or the product of two ten-digit primes, usually takes well under a second. If a factor is too large to find in time it is left as is and marked (not factored), so the page never freezes.

随机因数是怎样拆的？How does Random factors work?

先求出全部质因数，打乱后随机分成至少两组，每组相乘就是一个因数，所以 360 可能得到 4 x 90 或 6 x 6 x 10。质数无法拆开，会原样输出并说明。相同种子结果相同。All prime factors are found, shuffled and split at random into at least two groups; each group's product is one factor, so 360 may become 4 x 90 or 6 x 6 x 10. A prime cannot be split and is shown as is with a note. The same seed gives the same split.

「把整数改写成若干数之积」免费吗？需要注册吗？Is the Rewrite Integer as Product free? Do I need an account?

「把整数改写成若干数之积」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 Rewrite Integer as Product is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.