Interesting numbers...

Everyone has heard of the fact that 1729 is interesting because it is the smallest positive integer expressible as the sum of two cubes in two ways:

1729 = 9^3 + 10^3
1729 = 1^3 + 12^3

Can anyone think of the corresponding number for biquadrates? It is certainly very large. If not, can anyone give a tidy estimate of its order of magnitude?

Sorry, no idea on the biquadrates...

But if anyone could tell me something interesting about the number 51 I would be HUGELY grateful. It's part of a bet I made with someone where I said that every number had something interesting about it after we started talking about the number 26 being the only number that lies between a square and a cube.

I did have something on 51, but didn't write it down and now can't remember what it was and can't work it out 🙁 Anyone?

1. Mozart's extremely cool "La Finta Semplice" is listed as K. 51

2. Its binary representation 110011 looks kind of cool (palindrome et al)

3. It is the sum of a square and a prime 4 + 47 = 51

4. It is the sum of a cube and a prime 8 + 43 = 51

5. It is the difference of a triangle and a square 51 = 55 - 4

6. It is the sum of two trianlges 51 = 45 + 6

Does that do it?

After wasting the last ten minutes of my life using google I have found what I was looking for:-

"51

51 is the the first "boring" number according to David Wells' book "Curious and Interesting Numbers", i.e. it is the first positive integer not to appear.

Actually, an interesting thing about 51 is that it is the first odd composite number not to appear in times tables (which traditionally went up to 12 times), which means that it is probably the first number that many people would fail to recognise as being composite."

Taken from http://www.1729.com/topics/51.html

Come to think of it though, the very fact that it's the first "boring" positive integer to appear in that book makes it highly fascinating!

Originally posted by royalchicken

Can anyone think of the corresponding number for biquadrates? It is certainly very large.

I can't, but my computer can.

59^4 + 158^4 = 635318657
133^4 + 134^4 = 635318657

Originally posted by BigDoggProblem
I can't, but my computer can.

59^4 + 158^4 = 635318657
133^4 + 134^4 = 635318657

Right on. That one was a long time in coming; well done 😀!

Originally posted by royalchicken
Right on. That one was a long time in coming; well done 😀!

lol, I didn't realize how old the thread was! I will have to give the assist to mmhare for bringing the thread up again.