Primality Testing — A Deterministic Algorithm

Abstract

The most ancient and simple method for testing if a number is prime or not consists in factoring n. Using the fact that a non prime has a divisor r such that EquationSource<math display='block'> <mrow> <mn>1</mn><mo>&#x2264;</mo><mi>r</mi><mo>&#x2264;</mo><msqrt> <mi>n</mi> </msqrt> </mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$1 \leqslant r \leqslant \sqrt n$$. We obtain a method usable for numbers up to 10¹⁶ on a computer. It necessitates EquationSource<math display='block'> <mrow> <mn>0</mn><mrow><mo>(</mo> <mrow> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo>)</mo></mrow></mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$0\left( {\sqrt n } \right)$$ operations, which is large when n is big. Remarkable improvements have been made. On a pocket calculator one can factor numbers up to 10¹⁸ and on a large computer up to 10⁴⁰ with 0(n^1/4) algorithms.