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 re: eureca ! ! 308 in five fives by pcbouhid)

pcbouhid,

Looking back at the earlier posts I see that the termial function is acceptable, thus I have found a solution for the elusive 1037.
The name and symbol appears to have been first proposed (if you can call it a proposal) by Donald Knuth in 1973 in "The Art of Computer Programming". Both the name and notation has been repeated by Donald Sannella in his own 1994 book "Programming Languages and Systems - ESOP '94". As an example of the notation:
5? = 1+2+3+4+5.
Peter Luschny proposed another name and another notation for this 'function' of the triangular numbers in an OEIS Wiki article.
He called the function an 'addorial'.
5⁺ = 5+4+3+2+1
Luschny also introduced the notations for 'multi-addorials', analogous to the multifactorials:
5⁺⁺ = 5⁺⁽²⁾ = 5+3+1 = 9
5⁺⁺⁺ = 5⁺⁽³⁾ = 5+2 = 7
etc.

Another notation that could be acceptable is using the wreath product. As the special character is not easily noted in most fonts, I shall denote it here as a tilde (in appearance, a wreath product symbol turned 90 degrees). The symbol is used like the exclamation point for the factorial, but for the 'swinging factorial', and a pair of them, like the double factorial, for the 'double swinging factorial'. (The values of the odd indexes of the double swinging factorial are the same as for the double factorial, thus 5~~ = 5!! = 15.  By coincidence, 5~ = 5# = 30, yet other values can be achieved using the notation, e.g., (5+5)~ = 252.)

 

Edited on February 7, 2013, 1:43 am

Posted by Dej Mar on 2012-12-24 09:39:21