A mathematician who was exceedingly fond of the number five set to work trying to express as many consecutive integers using no numerals besides '5', and only up to five of them. She allowed herself to use any standard mathematical notation she knew, as long as it didn't contain any numerals. For example, she could use the symbol for 'square root', but not 'cube root' (because it contains a '3'). She determined that the highest consecutive integer she could express this way was 36. Her last few calculations were as follows:

• 31 = 5*5 + 5 + (5/5)
• 32 = 55*.5 + 5 - .5
• 33 = (55 + 5) * .55
• 34 = 5!/5 + 5/.5
• 35 = (5 + (5+5)/5) * 5
• 36 = 5*5 + 55/5
• 37 = ?

Was she correct in thinking 36 was the highest consecutive integer she could express this way? Can you express 37 using only up to five 5's?

Note: The intention here is to find an exact expression, so rounding expressions like [] "greatest integer" are not allowed.
Note: Can you do it without using letters of any kind (x, log, lim, sum, etc.)?

(In reply to eureca ! ! 308 in five fives by Ady TZIDON)

I accept this, Ady!!!

I was reading about the Gamma function that is represented by a symbol (a greek letter) that we are unable to show here. And I think that it is acceptable too, since itīs a standard math notation, like factorial (!).

Only for purpose of showing it (for those who donīt know yet), and using G, we have G(n) = (n-1)! and so G(5) = 24. (24 with only one five).

And, why disregard 5? = 1 + 2 + 3 + 4 + 5, the termial of 5, also a standard math notation, like "!"?

!The doors remains open! 

Edited on October 13, 2005, 9:08 pm

Edited on October 13, 2005, 9:09 pm

Posted by pcbouhid on 2005-10-13 21:02:31