An Efficient Way to Quickly Find All the Odd Prime Numbers Less Than 1369

As I’ve used different sized grids to play with the Sieve of Eratosthenes, I’ve decided I like one size grid better than all the rest. It has six odd numbers across, but I repeat the first column on the right because of convenience. You already know the only even prime number is 2, so this grid can help you find all the rest of the primes up to 1369 = 37².

Look at the grid. What are some things that you notice about it?

Square numbers are never prime, so why do I have them outlined on the grid? Why are some of them crossed out? Is there a pattern for that, too?

If you’ve done a sieve where you cross out all the multiples of the prime numbers in order, perhaps you’ve noticed that the first multiple to get crossed out that hasn’t been crossed out before is always the prime number squared.

Therefore, don’t start with the prime numbers. Start with their squares! The squares of each of the prime numbers and the next five odd multiples after those squares are listed in a box on the left of the paper. Put a dot in the corner of each of those multiples. Recognize the pattern they make and strike through those numbers with a colored pencil. A ruler will be helpful. Continue the same pattern down to the bottom of the grid. Then do the same thing with the next square of a prime number.  I’ve made a gif of these instructions being applied to a much smaller grid.
Finding Primes Less Than 361

make science GIFs like this at MakeaGif

It feels like I’m wrapping twinkling lights around a long sheet of cardboard!

Do try it on this much longer grid that goes to 1369. You’ll probably want to cut it out and glue or tape it together on the back. If this is an assignment, don’t cut off your name.

Read the following AFTER you’ve tried using the grid. I don’t want to spoil your sense of discovery!

To me, the lines drawn have a slope even if the lines are broken lines.
The slope for the 3s is undefined.
For the 5s, it’s +1; for the 7’s, it’s -1;
For the 11’s, it’s +2; for the 13’s, it’s -2;
For the 17’s, it’s +3; for the 19’s, it’s -3;
For the 23’s, it’s +4; for the 25’s, it’s -4; (You can skip the 25’s because they are already crossed out.)
For the 29’s, it’s +5; for the 31’s, it’s -5;
Cross out 37², and then you are done.

If the grid were longer, you could continue with the same pattern for as long as you want. I think it is pretty cool.

Now I’ll tell you a little bit about the number 1369:

  • 1369 is a composite number and is a perfect square.
  • Prime factorization: 1369 = 37 × 37 which can be written 1369 = 37²
  • 1369 has at least one exponent greater than 1 in its prime factorization so √1369 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1369 = (√37)(√37) = 37
  • The exponent in the prime factorization is 2. Adding one to that exponent, we get (2 + 1) = 3. Therefore 1369 has exactly 3 factors.
  • The factors of 1369 are outlined with their factor pair partners in the graphic below.

1369 is the sum of two squares:
35² + 12² = 1369

1369 is the hypotenuse of two Pythagorean triples:
444-1295-1369 which is (12-35-37) times 37
840-1081-1369 calculated from 2(35)(12), 35² – 12², 35² + 12²