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
The intention here is to find an exact expression, so rounding expressions like  "greatest integer" are not allowed.
Can you do it without using letters of any kind (x, log, lim, sum, etc.)?
(In reply to re(2): eureca ! ! 308 in five fives
by Ady TZIDON)
Strictly speaking, the gamma function does use a letter, so i have to vote against it.
As for subfactoial, it seems standard enough notation. I see no problem using it.
Maybe its the absolutist in me, but the idea of allowing notation on a 'one time use' basis makes me cringe. Either the notation should be allowed in all cases or it shouldn't.
well done, ady!