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.)?

I have been able to accomplish an impressive feat (i think). I have been able to find a way to make the number 779. I use primorial (#). You multiply all prime numbers up to and including x, and the product is x#. Using 5#, you can get 30, and below is the rest.

1. 5# = 30 five number 1
2. 30/5 = 6 five number 2
3. 6!=720 still 2 fives
4. 720 + (5# - .5)/.5 = 720 + 29.5/.5 = 720 + 59 = 779 five numbers 3, 4, and 5.

Thus 779 is possible, using a previously overlooked function.

Q.E.D.

Edited on February 3, 2006, 9:45 pm

Posted by Justin on 2006-02-03 21:21:02