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. Passing proves only this modular congruence, not that $n$ is a prime number. An odd composite number that passes a coprime base is a Fermat pseudoprime to that base. A Carmichael number passes every coprime base, so even testing all the units would not certify that

n

is a prime number. ^[ref]

This small table uses only the bases

2, 3, 5, 11

, each reduced modulo

n

first. A numeric entry is

a^{n - 1} mod n

: one passes, while any other numeric entry gives a Fermat witness. A "factor" entry records a proper greatest common divisor and stops before modular exponentiation. The prime/composite classification is independently established by trial division or the displayed prime factorization, not by the passing entries. These four bases are illustrative, with no asserted universal coverage or error probability.

Table 1.

Finite-base Fermat primality test examples with explicit greatest common divisor checks

$n$	Independent classification	$a = 2$	$a = 3$	$a = 5$	$a = 11$
2	Prime number	Zero residue: excluded	1	1	1
7	Prime number	1	1	1	1
15	$3 \cdot 5$ : composite number	4	Factor 3	Factor 5	1
341	$11 \cdot 31$ : composite number	1	56	67	Factor 11
561	$3 \cdot 11 \cdot 17$ : composite number	1	Factor 3	1	Factor 11

For

341 = 11 \cdot 31

g cd (2, 341) = 1

and

2^{10} = 1024 = 3 \cdot 341 + 1

, so

2^{340} = (2^{10})^{34} \equiv 1 (mod 341)

. Yet

g cd (3, 341) = 1

and

3^{340} \equiv 56 (mod 341)

, so three is a Fermat witness. One coprime base passes and another fails: being a Fermat pseudoprime depends on the base. Choosing eleven instead reveals the proper factor through

g cd (11, 341) = 11

For

561 = 3 \cdot 11 \cdot 17

, every coprime base passes: each of

2, 10, 16

divides

560

, so Fermat's little theorem gives

a^{560} \equiv 1

modulo each prime factor; the Chinese remainder theorem combines the three modular congruences. Equivalently, the Carmichael function gives

λ (561) = 80 ∣ 560

. Bases two and five have greatest common divisor one and pass, but

g cd (3, 561) = 3

g cd (11, 561) = 11

and

g cd (17, 561) = 17

expose factors. The qualification "coprime" is therefore crucial even for a Carmichael number.

The Fermat primality test is useful because a Fermat witness can prove compositeness using repeated squaring without finding a factor. Its failure to certify a passing candidate motivates stronger primality testing methods such as the Miller-Rabin primality test.

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Integer factorization Created 2026-10-13 Updated 2026-10-20

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Integer factorization seeks the prime factorization of a positive integer

N

. For

N > 1

, this means identifying its prime factors and their multiplicities; one has the empty prime factorization. Finding one proper divisor

1 < d < N

is useful progress, but need not finish the job: factor

d

and

N / d

further until all factors are prime numbers.

For example,

77 = 7 \cdot 11

is a semiprime, whereas

231 = 3 \cdot 7 \cdot 11

has three prime factors. Primality testing asks whether a proposed integer is prime; it need not return a factor when the answer is no. Trial division is a direct factoring method: try division by small prime numbers until a divisor is found. More elaborate methods obtain a proper divisor through a greatest common divisor of

N

with a specially constructed integer.

The beautiful trick in congruence of squares is that two integers with equal squares modulo

N

can reveal different factors through their sum and difference. In the worked

N = 77

example,

2 0^{2} \equiv 1 3^{2} (mod 77)

gives

g cd (20 - 13, 77) = 7

and

g cd (20 + 13, 77) = 11

. Computing those greatest common divisors uses the Euclidean algorithm; it does not require trying every potential factor. A factor base supplies a systematic way to construct useful equal-square relations. ^[ref].

This connection is useful in cryptography because recovering the factors of an RSA cryptosystem modulus defeats the factoring assumption behind its construction. That application concerns large, carefully chosen inputs; the small examples here explain the arithmetic mechanism and are not benchmarks or estimates of present cryptographic security.

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Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2025 / Paper 1 / 1 / b / Solution 2026-10-20

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It performs a trial division test for a prime number. An integer below

2

is rejected; otherwise the function searches for a divisor from

2

through

⌊ n ⌋

. A composite number has a prime factor no larger than its square root, so finding no such divisor proves primality. For example, mystery2(97) returns True, while mystery2(91) returns False, since

91 = 7 \cdot 13

This describes the intended mathematical test. For extremely large integers, the floating-point computation n**0.5 can round or overflow; an implementation intended for arbitrary-size integers should use an exact integer square root.

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Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Imperative programming / 5 / c / Solution 2026-10-31

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Use the iterator adapter from part (b) with a primality test by trial division. A composite positive integer has a divisor no greater than its square root.

def primeIter(upperBound: Int): Iter[Int] = {
  def isPrime(n: Int): Boolean = {
    if (n < 2) false
    else {
      var d = 2
      while (d <= n / d && n % d != 0) d += 1
      d > n / d
    }
  }
  filterIter[Int](isPrime _, new RangeIter(2, upperBound, 1))
}

Before each trial division iteration, no integer in

[2, d)

divides n. If division succeeds, the condition becomes false with

d \leq n / d

, and isPrime returns false. Otherwise the loop ends with

d > n / d

, so every possible small divisor has been excluded. The quotient comparison is equivalent to

d^{2} \leq n

for positive values and avoids integer overflow in d*d.

primeIter (23) : 2, 3, 5, 7, 11, 13, 17, 19.

(1)

Bounds at most two give an empty iterator. This method tests candidates lazily and uses

O (n)

trial divisions for candidate

n

; no efficiency requirement in the question forces a sieve of Eratosthenes implementation.

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Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Imperative programming / 2 / e / Solution 2026-10-31

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Use trial division on a shrinking residual. Once the trial divisor exceeds its square root, any residual larger than one is itself prime.

PROCEDURE divisors(n: INTEGER): INTEGER;
VAR residual, trial, exponent, answer: INTEGER;
BEGIN
  residual := n; trial := 2; answer := 1;
  WHILE trial * trial <= residual DO
    exponent := 0;
    WHILE residual MOD trial = 0 DO
      residual := residual DIV trial;
      exponent := exponent + 1
    END;
    answer := answer * (exponent + 1);
    trial := trial + 1
  END;
  IF residual > 1 THEN answer := 2 * answer END;
  RETURN answer
END divisors;

For the outer loop invariant, let

P

be the set of primes below trial and

e_{p}

their exponents in the original

n

. Then

n = residual \prod_{p \in P} p^{e_{p}}, answer = \prod_{p \in P} (e_{p} + 1),

(1)

and residual has no prime factor below trial. If trial divides the residual it must therefore be prime: a composite trial value would have a smaller prime factor, contradicting this clause. If it does not divide, the zero exponent contributes the factor

1

For the inner loop invariant, let

r_{0}

be the residual on entry. Then

r_{0} = trial^{exponent} residual

. Each exact division reduces the positive residual by at least a factor of two, so this loop terminates. At exit the exponent is the full multiplicity of trial; multiplying by exponent+1 preserves the outer loop invariant. The outer loop terminates because trial increases and residual never increases. When it exits, a composite residual would have a prime factor at most its square root, yet every smaller trial factor has already been excluded. Hence the residual is

1

or a single remaining prime of exponent

1

The procedure computes $τ (n)$ in $O (n)$ unit-cost time and $O (1)$ auxiliary space, with the separate successful divisions bounded by

⌊ lo g_{2} n ⌋

. No array or table of primes is needed, and

n = 1

correctly returns

1

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Prime number Created 2026-09-24 Updated 2026-10-13

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A prime number is a positive integer

p > 1

whose only positive divisors are

1

and

p

. The first ten prime numbers are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

. In contrast,

12 = 3 \cdot 4

is a composite number because it has a positive divisor strictly between one and itself. The integer one is neither a prime number nor a composite number: the lower bound

p > 1

is part of the definition. Equivalently, the divisor function satisfies

τ (p) = 2

for a prime number, whereas

τ (1) = 1

and

τ (n) \geq 3

for every composite number

n

Prime numbers are the building blocks of positive integers under multiplication. The fundamental theorem of arithmetic says that every integer greater than one is a product of prime numbers, with a unique collection of factors when their order is ignored. For example,

12 = 2 \cdot 2 \cdot 3

and

18 = 2 \cdot 3 \cdot 3

. Their common factors reveal the greatest common divisor

6

, and the combined maximum powers reveal the least common multiple

36

. A complicated divisibility question becomes a comparison of prime exponents.

There is also a useful finite test. If a composite number

n

equals

ab

with

1 < a \leq b

, then

a^{2} \leq n

. Therefore some prime factor is at most

n

, and trial division only needs to check prime numbers up to that square root. For

97

, checking

2, 3, 5, 7

suffices; none divides

97

, so it is a prime number. For

91

, the same test finds

7

, giving

91 = 7 \cdot 13

. To find many prime numbers together, the sieve of Eratosthenes reuses the work by marking multiples.

The division with remainder calculations for

97

make the test explicit:

97 = 2 \cdot 48 + 1 = 3 \cdot 32 + 1 = 5 \cdot 19 + 2 = 7 \cdot 13 + 6.

(1)

Since

9^{2} < 97 < 1 0^{2}

, these are all the prime numbers at most its square root, and every remainder is nonzero. Thus the calculation certifies a prime number, without testing the integers

8, 9, 10, \dots, 96

individually. In contrast,

91 = 7 \cdot 13 + 0

gives a prime factor and proves that

91

is a composite number.

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Trial division 2026-10-14

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Trial division searches for divisors of an integer by testing candidate divisions. For primality testing of an integer

n > 1

, testing the prime numbers up to

n

suffices: any composite number has a prime factor in that range. After finding a prime factor, repeated division removes its prime power, and testing the remaining quotient gives an elementary integer factorization method. This is useful for small inputs, but the number of candidates can grow rapidly with the input's digit count.

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