How to Identify a Prime Number without a Computer

"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 Edouard 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."

The 39-digit number 170,141,183,460,469,231,731,687,303,715,884,105,727 was proven prime by Edouard Lucas using a manual method from the 1800s. Prime numbers are divisible only by 1 and themselves; composite integers factor uniquely into primes. Primes appear with regularity and unpredictable fluctuations. The largest known prime (as of October 2025) has 41,024,320 digits. Many record primes are Mersenne primes of the form 2^p−1 with prime p; Lucas's number is the Mersenne prime 2^127−1, though not every number of that form is prime. Lucas spent almost two decades working on that proof and developed techniques that remain useful in modern primality testing.

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