Blog Post No.95: Extending Pythagoras pairs

Pythagoras theorem doesn't need an introduction and has lot of relevance in day to day applications and not just a mere theorem

Just made an attempt to look at some of the existing Pythagoras pairs to see if they throw some interesting patterns when we bring a cube into picture

Here you go...

a) 12 ^ 2 + 5 ^ 2 = 169 which is 13 ^ 2. Now add 3 ^ 3 to 13 ^ 2 --> 13 ^ 2 + 3 ^ 3 = 196 which is 14 ^ 2

b) 24 ^ 2 + 7 ^ 2 = 625 which is 25 ^ 2. Now add 6 ^ 3 to 25 ^ 2 --> 25 ^ 2 + 6 ^ 3 = 841 which is 29 ^ 2

Can someone figure out more patterns of this nature?

Blog Post 94: Interesting patterns

These are few patterns which may be obvious for Maths enthusiasts and also there is a formulae based proof but neverthless pretty fascinating when we see how these play out

a) 111 x 111 = 12321

11 + 1 = 12 and 12x12 = 144 which is 123+21

b) 112 x 112 = 12544

11+2 = 13 and 13x13 = 169 which is 125+44. Not only that this number is more unique for the following reasons

11-2 = 9 and 9x9 = 81 which is 125-44! (we don't see that in the earlier example due to a very standard formulae based logic

c) 113x 113 = 12769

11+3 = 14 and 14x14 = 196 which is 127+69

d) 114x114 = 12996

11+4 = 15 and 15x15 = 225 which is 129+96

This pattern doesn't repeat when we go the series starting with 12 or 13. Can someone figure out why this is happening?