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 the following, a number preceded by a raised dagger (^†) or daggers indicates a reference. The number of daggers indicate the number of 5's used to form the number. The first list is provided simply for reference, some with a note indicating the type of symbol notation used. The second list is the list of numbers 1001 to 1036. It is possible to generate higher sequences, yet I have not yet discovered how to form 1037.

Take note that a limit is now imposed that restricts the number of any distinct symbols to 5, just as the limit has been for the number of fives. Though the following conforms to those restrictions, it is not readily seen in the following.

  1/2   .5
  5/9   .5... --- non-terminating decimal
    1   5 / 5
    2   ^†30 / ^†15
    3   ^†15 / 5
    4   ^†120 / ^†30
    6   ^†30 / 5
    8   ^†120 / ^†15
    9   5 / .5... --- non-terminating decimal {.n...}
   10   5!!! --- multifactorial, 3rd degree {n!!!}
   12   (^††3)$ --- superfactorial {n$}
   15   5!! --- double factorial {n!!}
   30   5# --- primorial {n#}
   44   !5 --- subfactorial {!n}
  120   5! --- factorial {n!}
  900   ^†30 * ^†30
  945   (^††9)!!
15120   <5>₅ --- rising factorial {<x>_n}
===============================================
 
1001   ^††945 + ^†44 + ^††12
1002   ^†††1008 - ^††6
1003   ^†††1008 - 5
1004   ^†††1008 - ^††4
1005   ^†††1008 - ^††3
1006   ^†††1008 - ^††2
1007   ^†††1008 - ^††1
1008   ^††15120 / ^†15
1009   ^†††1008 + ^††1
1010   ^†††1008 + ^††2
1011   ^†††1008 + ^††3
1012   ^†††1008 + ^††4
1013   ^†††1008 + 5
1014   ^†††1008 + ^††6
1015   ^†††1020 - 5
1016   ^†††1020 - ^††4
1017   ^†††1020 - ^††3
1018   ^†††1020 - ^††2
1019   ^†††1024 - 5
1020   ^††900 + ^†120
1021   ^†††1020 + ^††1
1022   ^†††1020 + ^††2
1023   ^†††1020 + ^††3
1024   (^††4)⁵
1025   ^†††1024 + ^††1
1026   ^†††1024 + ^††2
1027   ^†††1024 + ^††3
1028   ^†††1024 + ^††4
1029   ^†††1024 + 5
1030   ^†††1020 + ^†10
1031   (^†120 * ^††9) - ^†44 - 5
1032   (^†120 + ^††9) * ^††8
1033   ^†††1008 + ^†30 - 5
1034   ^†††1020 + ^†44 - ^†30
1035   ^†††1020 + ^†15
1036   ^†††1024 + ^††12
1037   ?

 

Edited on December 16, 2012, 2:37 am

Posted by Dej Mar on 2012-12-16 02:06:11