Methods of computing square roots

In numerical analysis, a branch of mathematics, there are several square root algorithms or methods of computing the principal square root of a non-negative real number. For the square roots of a negative or complex number, see below.
Finding ${\displaystyle {\sqrt {S}}}$ is the same as solving the equation ${\displaystyle f(x)=x^{2}-S=0\,\!}$ for a positive ${\displaystyle x}$. Therefore, any general numerical root-finding algorithm can be used. Newton's method, for example, reduces in this case to the so-called Babylonian method:

${\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}=x_{n}-{\frac {x_{n}^{2}-S}{2x_{n}}}={\frac {1}{2}}\left(x_{n}+{\frac {S}{x_{n}}}\right)}$

These methods generally yield approximate results, but can be made arbitrarily precise by increasing the number of calculation steps.

Contents

• 1 Rough estimation
• 2 Babylonian method
  • 2.1 Example
  • 2.2 Convergence
    • 2.2.1 Worst case for convergence
• 3 Digit-by-digit calculation
  • 3.1 Basic principle
  • 3.2 Decimal (base 10)
    • 3.2.1 Examples
  • 3.3 Binary numeral system (base 2)
    • 3.3.1 Example
• 4 Exponential identity
• 5 Bakhshali approximation
  • 5.1 Example
• 6 Vedic duplex method for extracting a square root
  • 6.1 Basic principle
  • 6.2 Example
• 7 A two-variable iterative method
• 8 Iterative methods for reciprocal square roots
  • 8.1 Goldschmidt’s algorithm
• 9 Taylor series
• 10 Other methods
• 11 Continued fraction expansion
  • 11.1 Algorithm
  • 11.2 Example, square root of 114 as a continued fraction
  • 11.3 Generalized continued fraction
• 12 Using Pell's equation
• 13 Approximations that depend on the floating point representation
  • 13.1 Reciprocal of the square root
• 14 Negative or complex square
• 15 See also
• 16 Notes
• 17 External links

Primality test

A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might be called compositeness tests instead of primality tests.

Contents

• 1 Simple methods
  • 1.1 Pseudocode
• 2 Heuristic tests
• 3 Probabilistic tests
  • 3.1 Fermat primality test
  • 3.2 Miller–Rabin and Solovay–Strassen primality test
  • 3.3 Frobenius primality test
  • 3.4 Other tests
• 4 Fast deterministic tests
• 5 Complexity
• 6 Number-theoretic methods
• 7 References
• 8 Sources
• 9 External links

Simple methods

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The simplest primality test is trial division: Given an input number n, check whether any prime integer m from 2 to √n evenly divides n (the division leaves no remainder). If n is divisible by any m then n is composite, otherwise it is prime.^[1]
For example, we can do a trial division to test the primality of 100. Let's look at all the divisors of 100:

2, 4, 5, 10, 20, 25, 50

Here we see that the largest factor is 100/2 = 50. This is true for all n: all divisors are less than or equal to n/2. If we take a closer look at the divisors, we will see that some of them are redundant. If we write the list differently:

100 = 2 × 50 = 4 × 25 = 5 × 20 = 10 × 10 = 20 × 5 = 25 × 4 = 50 × 2

the redundancy becomes obvious. Once we reach 10, which is √100, the divisors just flip around and repeat. Therefore, we can further eliminate testing divisors greater than √n.^[1] We can also eliminate all the even numbers greater than 2, since if an even number can divide n, so can 2.
Let's look at another example, and use trial division to test the primality of 17. Since we now know we do not need to test using divisors greater than √n, we only need to use integer divisors less than or equal to ${\displaystyle \scriptstyle {\sqrt {17}}\approx 4.12}$. Those would be 2, 3, and 4. As stated above, we can skip 4 because if 4 evenly divides 17, 2 must also evenly divide 17, which we already would have checked before that. That leaves us with just 2 and 3. After dividing, we find that 17 is not divisible by 2 or 3, and we can confirm that 17 must be prime.
The algorithm can be improved further by observing that all primes are of the form 6k ± 1, with the exception of 2 and 3. This is because all integers can be expressed as (6k + i) for some integer k and for i = −1, 0, 1, 2, 3, or 4; 2 divides (6k + 0), (6k + 2), (6k + 4); and 3 divides (6k + 3). So, a more efficient method is to test if n is divisible by 2 or 3, then to check through all the numbers of form ${\displaystyle \scriptstyle 6k\ \pm \ 1\leq {\sqrt {n}}}$. This is 3 times as fast as testing all m.
Generalising further, it can be seen that all primes are of the form c#k + i for i < c# where i represents the numbers that are coprime to c# and where c and k are integers. For example, let c = 6. Then c# = 2 · 3 · 5  = 30. All integers are of the form 30k + i for i = 0, 1, 2,...,29 and k an integer. However, 2 divides 0, 2, 4,...,28 and 3 divides 0, 3, 6,...,27 and 5 divides 0, 5, 10,...,25. So all prime numbers are of the form 30k + i for i = 1, 7, 11, 13, 17, 19, 23, 29 (i.e. for i < 30 such that gcd(i,30) = 1). Note that if i and 30 are not coprime, then 30k + i is divisible by a prime divisor of 30, namely 2, 3 or 5, and is therefore not prime.
As c → ∞, the number of values that c#k + i can take over a certain range decreases, and so the time to test n decreases. For this method, it is also necessary to check for divisibility by all primes that are less than c. Observations analogous to the preceding can be applied recursively, giving the Sieve of Eratosthenes.
A good way to speed up these methods (and all the others mentioned below) is to pre-compute and store a list of all primes up to a certain bound, say all primes up to 200. (Such a list can be computed with the Sieve of Eratosthenes or by an algorithm that tests each incremental m against all known primes < √m). Then, before testing n for primality with a serious method, n can first be checked for divisibility by any prime from the list. If it is divisible by any of those numbers then it is composite, and any further tests can be skipped.
A simple, but very inefficient primality test uses Wilson's theorem, which states that p is prime if and only if:

${\displaystyle (p-1)!\equiv -1{\pmod {p}}\,}$

Although this method requires about p modular multiplications, rendering it impractical, theorems about primes and modular residues form the basis of many more practical methods.

Pseudocode

The following is a simple primality test in pseudocode for not very large numbers.

function is_prime(n)
    if n ≤ 1
        return false
    else if n ≤ 3
        return true
    else if n mod 2 = 0 or n mod 3 = 0
        return false
    let i ← 5
    while i * i ≤ n
        if n mod i = 0 or n mod (i + 2) = 0
            return false
        i ← i + 6
    return true

Heuristic tests

These are tests that seem to work well in practice, but are unproven and therefore are not, technically speaking, algorithms at all. The Fermat test and the Fibonacci test are simple examples, and they are very effective when combined. John Selfridge has conjectured that if p is an odd number, and p ≡ ±2 (mod 5), then p will be prime if both of the following hold:

• 2^p−1 ≡ 1 (mod p),
• f_p+1 ≡ 0 (mod p),

where f_k is the k-th Fibonacci number. The first condition is the Fermat primality test using base 2.
Selfridge, Carl Pomerance, and Samuel Wagstaff together offer $620 for a counterexample. The problem is still open as of September 11, 2015.^[2] Pomerance thinks that Selfridge's estate would probably pay his share of the reward.^[3]
The Baillie-PSW primality test is another excellent heuristic, using the Lucas sequence in place of the Fibonacci sequence. It has no known counterexamples.

Probabilistic tests

Probabilistic tests are more rigorous than heuristics in that they provide provable bounds on the probability of being fooled by a composite number. Many popular primality tests are probabilistic tests. These tests use, apart from the tested number n, some other numbers a which are chosen at random from some sample space; the usual randomized primality tests never report a prime number as composite, but it is possible for a composite number to be reported as prime. The probability of error can be reduced by repeating the test with several independently chosen values of a; for two commonly used tests, for any composite n at least half the a's detect n's compositeness, so k repetitions reduce the error probability to at most 2^−k, which can be made arbitrarily small by increasing k.
The basic structure of randomized primality tests is as follows:

1. Randomly pick a number a.
2. Check some equality (corresponding to the chosen test) involving a and the given number n. If the equality fails to hold true, then n is a composite number, a is known as a witness for the compositeness, and the test stops.
3. Repeat from step 1 until the required accuracy is achieved.

After one or more iterations, if n is not found to be a composite number, then it can be declared probably prime.

Fermat primality test

The simplest probabilistic primality test is the Fermat primality test (actually a compositeness test). It works as follows:

Given an integer n, choose some integer a coprime to n and calculate a^{n − 1} modulo n. If the result is different from 1, then n is composite. If it is 1, then n may or may not be prime.

If aⁿ⁻¹ (modulo n) is 1 but n is not prime, then n is called a pseudoprime to base a. In practice, we observe that, if aⁿ⁻¹ (modulo n) is 1, then n is usually prime. But here is a counterexample: if n = 341 and a = 2, then

${\displaystyle 2^{340}\equiv 1{\pmod {341}}}$

even though 341 = 11·31 is composite. In fact, 341 is the smallest pseudoprime base 2 (see Figure 1 of ^[4]).
There are only 21853 pseudoprimes base 2 that are less than 2.5×10¹⁰ (see page 1005 of ^[4]). This means that, for n up to 2.5×10¹⁰, if 2ⁿ⁻¹ (modulo n) equals 1, then n is prime, unless n is one of these 21853 pseudoprimes.
Some composite numbers (Carmichael numbers) have the property that a^{n − 1} is 1 (modulo n) for every a that is coprime to n. The smallest example is n = 561 = 3·11·17, for which a⁵⁶⁰ is 1 (modulo 561) for all a coprime to 561. Nevertheless, the Fermat test is often used if a rapid screening of numbers is needed, for instance in the key generation phase of the RSA public key cryptographic algorithm.

Miller–Rabin and Solovay–Strassen primality test

The Miller–Rabin primality test and Solovay–Strassen primality test are more sophisticated variants, which detect all composites (once again, this means: for every composite number n, at least 3/4 (Miller–Rabin) or 1/2 (Solovay–Strassen) of numbers a are witnesses of compositeness of n). These are also compositeness tests.
The Miller–Rabin primality test works as follows: Given an integer n, choose some positive integer a < n. Let 2^sd = n − 1, where d is odd. If

${\displaystyle a^{d}\not \equiv 1{\pmod {n}}}$

and

${\displaystyle a^{2^{r}d}\not \equiv -1{\pmod {n}}}$ for all ${\displaystyle 0\leq r\leq s-1,}$

then n is composite and a is a witness for the compositeness. Otherwise, n may or may not be prime. The Miller–Rabin test is a strong pseudoprime test (see PSW^[4] page 1004).
The Solovay–Strassen primality test uses another equality: Given an odd number n, choose some integer a < n, if

${\displaystyle a^{(n-1)/2}\not \equiv \left({\frac {a}{n}}\right){\pmod {n}}}$, where ${\displaystyle \left({\frac {a}{n}}\right)}$ is the Jacobi symbol,

then n is composite and a is a witness for the compositeness. Otherwise, n may or may not be prime. The Solovay–Strassen test is an Euler pseudoprime test (see PSW^[4] page 1003).
For each individual value of a, the Solovay–Strassen test is weaker than the Miller–Rabin test. For example, if n = 1905 and a = 2, then the Miller-Rabin test shows that n is composite, but the Solovay–Strassen test does not. This is because 1905 is an Euler pseudoprime base 2 but not a strong pseudoprime base 2 (this is illustrated in Figure 1 of PSW^[4]).

Frobenius primality test

The Miller–Rabin and the Solovay–Strassen primality tests are simple and are much faster than other general primality tests. One method of improving efficiency further in some cases is the Frobenius pseudoprimality test; a round of this test takes about three times as long as a round of Miller–Rabin, but achieves a probability bound comparable to seven rounds of Miller–Rabin.
The Frobenius test is a generalization of the Lucas pseudoprime test. One can also combine a Miller–Rabin type test with a Lucas pseudoprime test to get a primality test that has no known counterexamples. That is, this combined test has no known composite n for which the test reports that n is probably prime. One such test is the Baillie–PSW primality test, several variations of which exist.^[5]

Other tests

Leonard Adleman and Ming-Deh Huang presented an errorless (but expected polynomial-time) variant of the elliptic curve primality test. Unlike the other probabilistic tests, this algorithm produces a primality certificate, and thus can be used to prove that a number is prime.^[6] The algorithm is prohibitively slow in practice.
If quantum computers were available, primality could be tested asymptotically faster than by using classical computers. A combination of Shor's algorithm, an integer factorization method, with the Pocklington primality test could solve the problem in ${\displaystyle O(\log ^{3}n\log \log n\log \log \log n)}$.^[7]

Fast deterministic tests

Near the beginning of the 20th century, it was shown that a corollary of Fermat's little theorem could be used to test for primality.^[8] This resulted in the Pocklington primality test.^[9] However, as this test requires a partial factorization of n − 1 the running time was still quite slow in the worst case. The first deterministic primality test significantly faster than the naive methods was the cyclotomy test; its runtime can be proven to be O((log n)^{c log log log n}), where n is the number to test for primality and c is a constant independent of n. Many further improvements were made, but none could be proven to have polynomial running time. (Note that running time is measured in terms of the size of the input, which in this case is ~ log n, that being the number of bits needed to represent the number n.) The elliptic curve primality test can be proven to run in O((log n)⁶), if some conjectures on analytic number theory are true.^[which?] Similarly, under the generalized Riemann hypothesis, the deterministic Miller's test, which forms the basis of the probabilistic Miller–Rabin test, can be proved to run in Õ((log n)⁴).^[10] In practice, this algorithm is slower than the other two for sizes of numbers that can be dealt with at all. Because the implementation of these two methods is rather difficult and creates a risk of programming errors, slower but simpler tests are often preferred.
In 2002, the first provably unconditional deterministic polynomial time test for primality was invented by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. The AKS primality test runs in Õ((log n)¹²) (improved to Õ((log n)^7.5)^[11] in the published revision of their paper), which can be further reduced to Õ((log n)⁶) if the Sophie Germain conjecture is true.^[12] Subsequently, Lenstra and Pomerance presented a version of the test which runs in time Õ((log n)⁶) unconditionally.^[13]
Agrawal, Kayal and Saxena suggest a variant of their algorithm which would run in Õ((log n)³) if Agrawal's conjecture is true; however, a heuristic argument by Hendrik Lenstra and Carl Pomerance suggests that it is probably false.^[11] A modified version of the Agrawal's conjecture, the Agrawal–Popovych conjecture,^[14] may still be true.

Complexity

In computational complexity theory, the formal language corresponding to the prime numbers is denoted as PRIMES. It is easy to show that PRIMES is in Co-NP: its complement COMPOSITES is in NP because one can decide compositeness by nondeterministically guessing a factor.
In 1975, Vaughan Pratt showed that there existed a certificate for primality that was checkable in polynomial time, and thus that PRIMES was in NP, and therefore in NP ∩ coNP. See primality certificate for details.
The subsequent discovery of the Solovay–Strassen and Miller–Rabin algorithms put PRIMES in coRP. In 1992, the Adleman–Huang algorithm^[6] reduced the complexity to ZPP = RP ∩ coRP, which superseded Pratt's result.
The Adleman–Pomerance–Rumely primality test from 1983 put PRIMES in QP (quasi-polynomial time), which is not known to be comparable with the classes mentioned above.
Because of its tractability in practice, polynomial-time algorithms assuming the Riemann hypothesis, and other similar evidence, it was long suspected but not proven that primality could be solved in polynomial time. The existence of the AKS primality test finally settled this long-standing question and placed PRIMES in P. However, PRIMES is not known to be P-complete, and it is not known whether it lies in classes lying inside P such as NC or L. It is known that PRIMES is not in AC⁰.^[15]

Number-theoretic methods

Certain number-theoretic methods exist for testing whether a number is prime, such as the Lucas test and Proth's test. These tests typically require factorization of n + 1, n − 1, or a similar quantity, which means that they are not useful for general-purpose primality testing, but they are often quite powerful when the tested number n is known to have a special form.
The Lucas test relies on the fact that the multiplicative order of a number a modulo n is n − 1 for a prime n when a is a primitive root modulo n. If we can show a is primitive for n, we can show n is prime.