Prime Patterns: 1 more or 1 less than multiple of 6; Ulam Spiral; Prime Primes (April 1,2,6,7)

The children are excited about the prospect of the multiple $1,000,000 prizes offered to the mathematician who can find a pattern in prime numbers that would predict the next prime. Hopefully, this will be a lifelong pursuit.

However, there are a number of beautiful patterns in primes that do not predict the next prime number. After playing with different grids of numbers, I found an interesting pattern and had the children discover this with a number grid of 6 across. Using the Sieve of Eratosthenes, eliminating the multiples of 2, 3, 5, 7, 11, and 13, after circling the same, the children found an amazing discovery. Every prime number except for 2 and 3 is one more or one less than a multiple of 6. That does not mean that every number that is one more or one less than a multiple of 6 is prime, but it is true of every prime number.

I then introduced them to the Ulam Spiral. Stanislaw Ulam, a scientist on the Manhattan Project, was doodling during a boring lecture and created a spiral of numbers and then circled the primes. He found a pattern of diagonals that was astounding. He also found a horizontal line of numbers starting with 8, 9, 10, 27 and so on that would never have a prime number in its horizontal line. The students should use their list of primes from the first activity to circle the primes in the second activity.

Finally, the name of the division in my school that comprises the kindergarten, first and second grades is called the Primes Division. My students wondered how many of the children in that division had prime number values to their name. They used the Dollar Word scale to value their first and last names. The remarkable fact is that 10 of the 52 students had prime values. This is roughly 1 in 5 or 20% of the students. This coincides with the number of primes in the first 300 numbers they worked on last week (62/300 or roughly 1/5).

Mathlete children can do the same activity with the children's names in their class or division.

Answers to each of these activities are found in the second pdf.