How to identify a prime number without a computer

Summary

Is 170,141,183,460,469,231,731,687,303,715,884,105,727 prime? Before you ask the Internet for an answer, can you consider how you might answer that question without a computer or even a digital calculator?In the 1800s French mathematician Édouard Lucas spent years proving that this 39-digit number was indeed prime. How did he do it? Lucas, who incidentally also designed the entertaining game Tower of Hanoi, developed a method that’s still useful today, more than a century later.People have been fascinated by prime numbers for millennia. These numbers are divisible only by 1 and themselves, whereas every other integer can be uniquely expressed as the product of several prime numbers; for example, 15 = 3 × 5. Prime numbers essentially form the periodic table of mathematics. They also hold many secrets. They appear on the number line with a certain regularity, but their occurrence is characterized by fluctuations that cannot yet be quantified. This unpredictability has been a source of consternation for experts.On supporting science journalismIf you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.And math enthusiasts are constantly seeking new prime numbers. The current record (as of October 2025) for the largest prime is 2136,279,841 − 1, a number with 41,024,320 digits. Simply reading this number aloud would take approximately 240 days.Prime Numbers with a Special StructureAnyone who has observed the record-breaking prime numbers of recent years may have noticed that they mostly have a similar structure: 2p – 1 (where p is a prime number). Prime numbers of this form are called Mersenne primes. And the number to which Lucas dedicated almost two decades of his life is also a Mersenne prime, namely 2127 – 1. But there’s some trickiness to these Mersenne primes: not every 2p– 1 is a prime...