Primes and squares · Grades 7–9 · Unsolved

Legendre's Conjecture

Between any two perfect squares that sit next to each other, there’s always a prime.

Each strip runs from one square to the next. Orange dots are primes. Every strip has at least two so far.

Try it

Between 4 and 9: 5, 7Between 16 and 25: 17, 19, 23Between 100 and 121: five primes

In 1912 it made a famous list of four prime problems that seemed out of reach. All four are still unsolved.

We do know there’s always a prime between any number and its double.

How long is each strip? Can you predict the next one’s length?

Which strip has the fewest primes? Could one ever have none?

Detailed one-pager (PDF) History, why it’s unsolved, and more classroom ideas