While browsing through NUS’s Undergraduate Research Opportunities Programme in Science, I discovered an interesting project report published in year 2001. The report is titled “Perspectives in Mathematics and Art” by Kevin Heng Ser Guan (then an first year undergraduate, now a physics professor, I guess!).
Web Report- http://www.math.nus.edu.sg/aslaksen/projects/perspective/
Print-friendly .pdf version- http://www.math.nus.edu.sg/~mathelmr/projects/kh2-urops.pdf
Recently I realized the special properties of following numbers:
It is the only number which is both real and purely imaginary at same time.
It is sufficient to create all the counting numbers (a.k.a. natural numbers).
This is the maximum exponent, , for which has solution in natural numbers. This peculiar property leads to “Fermat’s Last Theorem”.
If you also have some special numbers in mind, please do share them below as comments.
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.