Primes: popular and lonely

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ulam While doodling in class, I made a 10 x 10 grid and filled it with numbers from 1 to 100. The motivations behind 10 x 10 grid was human bias towards the number 10.

New Doc 10_1

Then inspired by Ulam Spiral, I started creating paths (allowing diagonal, horizontal and vertical moves) starting from the smallest number. Following paths emerged:

  • 2→ 3 →13 → 23
  • 2 → 11
  • 7 → 17
  • 19 → 29
  • 31 → 41
  • 37 → 47
  • 43 → 53
  • 61 → 71
  • 73 → 83
  • 79 → 89

So, longest path is of length 4 and others are of length 2.

The number 2 is special one here, since it leads to two paths. I will call such primes, with more than one paths, popular primes.

Now, 5, 59, 67 and 97 don’t have any prime number neighbour. I will call such primes, with no neighbour, lonely primes.

I hope to create other b \times b grids filled with 1 to b^2 natural numbers written in base b. Then will try to identify such lonely and popular primes.

If you find this idea interesting, please help me to create such grids.

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7 responses »

  1. Pingback: الشعبية – المتفرّدة: فهم الأعداد الأولية. | فِي سِتَّةِ أَيَّامٍ

  2. Pingback: Popular-Lonely primes understood | Gaurish4Math

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