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Mathematics · pattern A, Numeric fields

Is it prime, and if not, what are its factors.

Enter a number to test for primality, with its full factorisation, divisor list and divisor sum where it turns out to be composite.

Inputs

The formula used

a prime has exactly two divisors: 1 and itself

Trial division only needs to reach the square root, because any factor above it pairs with one below.

Primality

Prime

997 has no divisors other than 1 and itself.

Prime factorisation
997
All divisors
1, 997
Number of divisors
2
Sum of divisors
998
Perfect square?
no

Worked steps

  1. 1divide by 997

    trial division by successive primes until the quotient is 1

Testing to the square root is enough.

If n has a factor larger than √n, its cofactor is smaller than √n and would already have been found. That is why checking a four-digit number by hand is quick, and why primality testing scales as well as it does.

Questions about prime number

Why is 1 not prime?
Because unique factorisation would fail — 6 could be written with any number of 1s. Excluding it keeps the theorem clean.
How many primes are there?
Infinitely many, proved by Euclid. They thin out but never stop, at roughly n ÷ ln n below n.
Why do primes matter in cryptography?
Multiplying two large primes is easy and factorising the product is hard. RSA depends entirely on that asymmetry.

More in Mathematics

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A calculator handles the arithmetic. It cannot teach you the method.

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Arithmetic runs in double-precision floating point, so results beyond about fifteen significant figures are not exact. Where a question wants an exact fraction or surd, keep the exact form rather than a decimal.