Big Maths Ideas · Beta

Prime patterns

Compare a number grid and an Ulam spiral, then investigate the highlighted primes.

Select a numbered cell. Blue cells are prime in both arrangements.

\text{prime: exactly two positive divisors}

Scroll the diagram sideways to see every label.

41 is prime.

Selected number: 41
Positive divisors: 1, 41

The idea behind the experiment

An Ulam spiral arranges the positive integers on a square spiral and marks primes. Diagonal patterns can be striking, but a finite pattern does not guarantee infinitely many prime values along a line. Changing the layout never changes whether a number is prime.

Wolfram’s prime spiral reference

Predict, test and explain

Step 1 of 3

Predict

Is 1 a prime number?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

Is 1 a prime number?

$\text{positive divisors of }1$

Hint

Test positive divisors.

Reveal answer

1 has only one positive divisor. A prime has exactly two.

$\{1\}$

Build an explanation

Hint 1

Test positive divisors.

Hint 2

A prime is greater than 1.

Hint 3

A visual pattern is not a guarantee that every later number is prime.

Worked example

This example uses fixed values, separate from the diagram controls.

Decide whether 9 is prime.

9 is divisible by 1 and 9.
It is also divisible by 3.
Having more than two divisors makes it composite.

Answer: 9 is composite: divisors 1, 3, 9.

Connect this idea

Watch out: Being indivisible by smaller positive integers is not enough to make 1 prime.