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.

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 grids filled with 1 to natural numbers written in base . 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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it is quite interesting.. you can also lookup on Green Tao theorem(http://www.math.udel.edu/~lazebnik/papers/deptnewsletterprimes.pdf)

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Thanks! This is very accessible exposition of Green-Tao Theorem.

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When you do b^2 it wont be prime, ever.

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Yes, like 10^2 = 100. I just want to preserve this pattern. of 1 to 100 🙂

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Oh sorry, I see now. I misunderstood!

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