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Subject: "Four 3's"     Previous Topic | Next Topic
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Conferences The CTK Exchange Middle school Topic #87
Reading Topic #87
Vladimir
Member since Jun-22-03
 Aug-24-03, 06:00 PM (EST)
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"Four 3's"
 
   LAST EDITED ON Sep-15-03 AT 11:39 AM (EST)
 
More four 3's than at https://www.cut-the-knot.org/arithmetic/funny/4_3.shtml

27 = 33!/3·3 = 36/3·3 = 32·3 = 9·3 

28 = 33 + 3/3 = 27 + 1
28 = (Ö((3!)!!) + 3!·Ö((3!)!!)/Ö3 = (Ö(6!!) + 6·Ö(6!!))/Ö3 = (Ö48 + 6·Ö48)/Ö3 = (4·Ö3 + 4·6·Ö3)/Ö3 = 4 + 24

29 = 33 + 3!/3 = 27 + 6/3 = 27 + 2

30 = 3!·3! - 3 - 3 = 6·65 - 6 = 36 - 6
30 = 33 + 3! - 3 = 27 + 6 - 3 = 27 + 3
30 = 3·3·3 + 3 = 27 + 3
30 = ((3!)!! + (3!)!! - 3!)/3 = (6!! + 6!! - 6)/3 = (48 + 48 - 6)/3 = (96 - 6)/3 = 90/3
30 = (3!)!! - 3!·Ö(3·3) = 6!! - 6·3 = 48 - 18
30 = (3!)3/3! - 3! = 63/6 - 6 = 216/6 - 6 = 36 - 6

31 = 33 - 3!/3 = 33 - 6/3 = 33 - 2
31 = ((3!)!! + (3!)!! - 3)/3 = (6!! + 6!! - 3)/3 = (48 + 48 - 3)/3 = (96 - 3)/3 = 93/3

32 = (3!)!!/3·3!/3 = 6!!/3·6/3 = 48/3·2 = 16·2
   
33 = 33/3·3 = 33·1
33 = 33 + 3 - 3 = 33 + 0
33 = 33 + 3 + 3 = 27 + 6
33 = 3!·3! - 3! + 3 = 6·6 - 6 + 3 = 36 - 3
33 = (3!)3!/3 - 3 = 66/3 - 3 = 62 - 3 = 36 - 3
33 = 3·Ö(3·(3!)!!) - 3 = 3·Ö(3·6!!) - 3 = 3·Ö(3·48) - 3 = 3·Ö144 - 3 = 3·12 - 3 = 36 - 3
33 = ((3!)!! + (3!)!! + 3)/3 = (6!! + 6!! + 3)/3 = (48 + 48 + 3)/3 = (96 + 3)/3 = 99/3
33 = (3!)3/3! - 3 = 63/6 - 3 = 216/6 - 3 = 36 - 3

34 = 33 + 3/3 = 33 + 1
34 = 3!·3! - 3!/3 = 6·6 - 6/3 = 36 - 2
34 = ((3!)!! + (3!)!! + 3!)/3 = (6!! 6!! + 6)/3 = (48 + 48 + 6)/3 = (96 + 6)/3 = 102/3
 
35 = 3!·3! - 3/3 = 6·6 - 1 = 36 - 1
35 = 33 + 3!/3 = 33 + 6/3 = 33 + 2
35 = ((3!)!! + (3!)!!)/3 + 3 = (6!! + 6!!)/3 + 3 = (48 + 48)/3 + 3 = 96/3 + 3 = 32 + 3

36 = 3!·3! + 3 - 3 = 6·6 + 0 = 6·6
36 = 33 + 3·3 = 27 + 9
36 = Ö(3!·3!·3!·3!) = Ö(6·6·6·6) = 6·6

37 = 3!·3! + 3/3 = 6·6 + 1 = 36 + 1
37 = (3!)!! - 33/3 = 6!! - 11 = 48 - 11

38 = 3!·3! + 3!/3 = 6·6 + 6/3 = 36 + 2

39 = 33 + 3 + 3 = 33 + 6
39 = 3!·3! + 3! - 3 = 6·6 + 6 - 3 = 36 + 6 - 3 = 36 + 3 
39 = (3!)!! - 33!/3 = 6!! - 36/3 = 48 - 32 = 48 - 9
39 = (3!)3!/3 + 3 = 66/3 + 3 = 62 + 33 = 36 + 3
39 = 3·Ö(3·(3!)!!) + 3 = 3·Ö(3·6!!) + 3 = 3·Ö(3·48) + 3 = 3·Ö144 + 3 = 3·12 + 3 = 36 + 3
39 = (3! + 3!)·3 + 3 = (6 + 6)·3 + 3 = 12·3 + 3 = 36 + 3
39 = (3!)3/3! + 3 = 63/6 + 3 = 216/6 + 3 = 36 + 3

40 = (3 + 3!/3)!/3 = (3 + 6/3)!/3 = (3 + 2)!/3 = 5!/3 = 120/3
40 = (3!)!! - 3! - 3!/3 = 6!! - 6 - 6/3 = 48 - 6 - 2 = 48 - 8
40 = (3!)!! - (3!/3)3 = 6!! - (6/3)3 = 48 - 23 = 48 - 8

41 = (3!)!! - 3! - 3/3 = 6!! - 6 - 1 = 48 - 7

42 = (3!)!! - 3·3 + 3 = 6!! - 9 + 3 = 48 - 6
42 = 3!·3! + 3 + 3 = 6·6 + 6 = 36 + 6
42 = 3!·(3! + 3/3) = 6·(6 + 1) = 6·7
42 = 33 + 3·3 = 33 + 9
42 = (3!)3/3! + 3! = 63/6 + 6 = 216/6 + 6 = 36 + 6

43 = (3!)!! - 3! + 3/3 = 6!! - 6 + 1 = 48 - 5
43 = (3!)!! - 3!/3 - 3 = 6!! - 6/3 - 3 = 48 - 2 - 3 = 48 - 5

44 = (3!)!! - 3! + 3!/3 = 6!! - 6 + 6/3 = 48 - 6 + 2 = 48 - 4
44 = (3!)!! - 3 - 3/3 = 6!! - 3 - 1 = 48 - 4

45 = (3!)!! + 3 - 3 - 3 = 6!! + 3 - 6 = 48 - 3
45 = 3·(3 + 3!/3)!! = 3·(3 + 6/3)!! = 3·(3 + 2)!! = 3·5!! = 3·15 

46 = (3!)!! - 3 + 3/3 = 6!! - 3 + 1 = 48 - 2
46 = (3 + 3)!! - 3!/3 = 6!! - 6/3 = 48 - 2
46 = (3!)!! - Ö(3 + 3/3) = 6!! - Ö(3 + 1) = 48 - Ö4 = 48 - 2
46 = (3!)!! - Ö(3! - 3!/3) = 6!! - Ö(6 - 6/3) = 48 - Ö4 = 48 - 2
 
47 = (3!)!! - 3 + 3!/3 = 6!! - 3 + 6/2 = 48 - 3 + 2 = 48 - 1
47 = (3!)!! - (3/3)3 = 6!! - 13 = 48 - 1

48 = (3!)!! + 3! - 3 - 3 = 6!! + 6 - 6 = 48 + 0
48 = ((3!)!! + (3!)!! + (3!)!!)/3 = (6!! + 6!! + 6!!)/3 = (48 + 48 + 48)/3
48 = 3·3·3! - 3! = 9·6 - 6 = 54 - 6 

49 = (3!)!! + 3 - 3!/3 = 6!! + 3 - 6/3 = 48 + 3 - 2 = 48 + 1
49 = (3!)!! + (3/3)3 = 6!! + 13 = 48 + 1

50 = (3!)!! + Ö(3 + 3/3) = 6!! + Ö(3 + 1) = 48 + Ö4 = 48 + 2
50 = (3!)!! + Ö(3! - 3!/3)= 6!! + Ö(6 - 6/3) = 48 + Ö(6 - 2) = 48 + Ö4 + 48 + 2
50 = (3!)!! + (3!)!!/3! - 3! = 6!! + 6!!/6 - 6 = 48 + 48/6 - 6 = 42 + 8
  
51 = (3!)!! + 3 + 3 - 3 = 6!! + 3 + 0 = 48 + 3
51 = (3!)!! + 3·3/3 = 6!! + 9/3 = 48 + 3
51 = 3·3·3! - 3 = 9·6 - 3 = 54 - 3
51 = 33·3 - (3!)!! = 99 - 6!! - 99 - 48

52 = (3!)!! + 3 + 3/3 = 6!! + 3 + 1 = 48 + 4
52 = (3!)!! + 3! - 3!/3 = 6!! + 6 - 6/3 = 48 + 6 - 2 = 54 - 2
 
53 = (3!)!! + 3 + 3!/3 = 6!! + 3 + 6/3 = 48 + 3 + 2 = 48 + 5
53 = (3!)!! + (3!)!!/3! - 3 = 6!! + 6!!/3 - 3 = 48 + 48/6 - 3 = 45 + 8
 
54 = (3!)!! + 3! + 3 - 3 = 6!! + 6 + 0 = 48 + 6
54 = (3!)!! + 3!·3!/3! = 6!! + 6·6/6 = 48 + 6
54 = 33·3!/3 = 27·6/3 = 27·2
54 = 33 + 33 = 27 + 27
54 = 3!·3! + 3·3! = 6·6 + 3·6 = 36 + 18

55 = (3!)!! + 3! + 3/3 = 6!! + 6 + 1 = 48 + 7

56 = (3!)!! + (3 + 3/3)!! = 6!! + (3 + 1)!! = 48 + 8
56 = (3!)!! + 3! + 3!/3 = 6!! + 6 + 6/3 = 48 + 6 + 2 = 48 + 8
56 = (3!)!! + (3!/3)3 = 6!! + (6/3)3 = 48 + 23 = 48 + 8

57 = (3!)!! + 33/3 = 6!! + 27/3 = 48 + 9
57 = (3!)!! + 3! + 3! - 3 = 6!! + 6 + 6 - 3 = 48 + 9
57 = 3·3·3! + 3 = 9·6 + 3 = 54 + 3

58 = (3!/3)3! - 3! = (6/3)6 - 6 = 26 - 6 = 64 - 6 

59 = (3!)!! + 33/3 = 6!! + 11 = 48 + 11
59 = (3!)!! + (3!)!!/3! + 3 = 6!! + 6!!/6 + 3 = 48 + 48/6 + 3 = 51 + 8
 
60 = (3!)!! + 3!·3!/3 = 6!! + 6·6/3 = 48 + 36/3 = 48 + 12
60 = (3!)!! + 3·3 + 3 = 6!! + 9 + 3 = 48 + 12
60 = (3!)!! + (3!)!! - 3!·3! = 6!! + 6!! - 6·6 = 48 + 48 - 36 = 96 - 36
60 = (3!)!! + 3·3! - 3! = 6!! + 3·6 - 6 = 48 + 18 - 6 = 48 + 12
60 = 3·3·3! + 3! = 9·6 + 6 = 54 + 6
60 = 33 + 33 = 33 + 27

61 = (3!/3)3! - 3 = (6/3))6 - 3 - 64 - 3
61 = (3!)!! + (3!)!!/3 - 3 = 6!! + 6!!/3 - 3 = 48 + 48/3 - 3 = 45 + 16
 
62 = (3!)!! + (3!)!!/3! + 3! = 6!! + 6!!/6 + 6 = 48 + 48/6 + 6 = 48 + 8 + 6 = 48 + 14
 
63 = (3!)!! + 3·3 + 3! = 6!! + 9 + 6 = 48 + 15
63 = (3!)!! + 3! + 3! + 3 = 6!! + 6 + 6 + 3 = 48 + 15
63 = 3!·3/.3 + 3 = 6·10 + 3 = 60 + 3
63 = 33 + 3!·3! = 27 + 6·6 = 27 + 36
63 = (33 - 3!)·3 = (27 - 6)·3 = 21·3 

64 = (3 + 3/3)3 = (3 + 1)3 = 43
64 = (3! - 3!/3)3 = (6 - 6/3)3 = (6 - 2)3 = 43
64 = ((3!)!!/3!)·((3!)!!/3!) = (6!!/6)·(6!!/6) = (48/6)·(48/6) = 8·8
64 = ((3!)!!/3!)3!/3 = (6!!/6)6/3 = (48/6)2 = 82 

65 - see cheating

66 = 33·3!/3 = 33·6/3 = 33·2 
66 = 33 + 33
66 = (3!)!! + 3! + 3! + 3! = 6!! + 6 + 6 + 6 = 48 + 18
66 = Ö(3·(3!)!!)·3! - 3! = Ö(3·6!!)·6 - 6 = Ö(3·48)·6 - 6 = Ö144·6 - 6 = 12·6 - 6 = 72 - 6

67 = (3!/3)3! + 3 = (6/3)6 + 3 = 26 + 3 = 64 + 3  
67 = (3!)!! + (3!)!!/3 + 3 = 6!! + 6!!/3 + 3 = 48 + 48/3 + 3 = 51 + 16
 
68 = ???

69 = (3!)!! + 3·3! + 3 = 6!! + 3·6 + 3 = 48 + 18 + 3 = 48 + 21  
69 = Ö(3·(3!)!!)·3! - 3 = Ö(3·6!!)·6 - 3 = Ö(3·48)·6 - 3 = Ö144·6 - 3 = 12·6 - 3 = 72 - 3
69 = (3!)!! + 33 - 3! = 33 + 3!·3! = 6!! + 27 - 6 = 48 + 21

70 = (3!/3)3! + 3! = (6/3)6 + 6 = 26 + 6 = 64 + 6 
70 = (3!)!! + (3!)!!/3 + 3! = 6!! + 6!!/3 + 6 = 48 + 48/3 + 6 = 54 + 16

71 - see cheating

72 = 3·3·(3!)!!/3! = 9·6!!/6 = 9·48/6 = 9·8
72 = (33 - 3)·3 = (27 - 3)·3 = 24·3
72 = (3!)!! + 33 - 3 = 6!! + 27 - 3 = 48 + 27 - 3 = 48 + 24
72 = 3!·3! + 3!·3! = 6·6 + 6·6 = 36 + 36
72 = (3!)3 - 3·(3!)!! = 6!)3 - 3·6!! = 216 - 3·48 = 216 - 144
72 = (3!)! - 3·(3!)3 = 6! - 3·63 = 720 - 3·216 = 720 - 648
72 = (3 + 3)·Ö(3·(3!)!!) = 6·Ö(3·6!!) = 6·Ö(3·48) = 6·Ö144 = 6·12

---------------------------------------------------------------------
Cheating with .3 (i.e., 0.3) means going outside of integers:

40 = 3!/3·3!/.3

42 = 33/.3 - (3!)!! = 27/.3 - 6!! = 90 - 48

47 = 33 + 3!/.3 = 27 + 6/.3 = 27 + 20

52 = (3!)!! + 3/.3 - 3! = 6!! + 10 - 6 = 48 + 4

53 = 33 + 3!/.3 = 33 + 6/.3 = 33 + 20

54 = 3!·3/.3 - 3! = 6·10 - 6 = 60 - 6

55 = (3!)!! + 3/.3 - 3 = 6!! + 10 - 3 = 48 + 7

56 = 3!·3! + 3!/.3= 6·6 + 6/.3 = 36 + 20

57 = 3!·3/.3 - 3 = 6·10 - 3 = 60 - 3

61 = (3!)!! + 3/.3 + 3 = 6!! + 10 + 3 = 48 + 13

62 = (3!)!! + 3!/.3 - 3! = 6!! + 6/.3 - 6 = 48 + 20 - 6 = 68 - 6
62 = 33/.3 - (3!)!! = 110 - 6!! = 110 - 48

63 = 3!·3/.3 + 3 = 6·10 + 3 = 60 + 3

64 = (3!)!! + 3/.3 + 3! = 6!! + 10 + 6 = 48 + 16
64 = (3!)!!/.3 - (3!)!! - (3!)!!= 6!!/.3 - 6!! - 6!! = 48/.3 - 96 = 160 - 96

65 = (3!)!! + 3!/.3 - 3 = 6!! + 6/.3 - 3 48 + 20 - 3 = 68 - 3

66 = 3·3!/.3 + 3! = 3·6/.3 + 6 = 3·20 + 6 = 60 + 6

71 = (3!)!! + 3!/.3 + 3 = 6!! + 6/.3 + 3 = 48 + 20 + 3 = 68 + 3

72 = 3!·3!/.3 - (3!)!! = 6·6/.3 - 6!! = 36/.3 - 48 = 120 - 48

---------------------------------------------------------------------
Cheating with the integer part operator (also called the floor function) such as {Ö(3!)} = 2 or {Ö(33)} = 5 is ugly .


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  Subject     Author     Message Date     ID  
  RE: Four 3's Vladimir Aug-24-03 1
     RE: Four 3's Vladimir Aug-24-03 2
  RE: Four 3's sfwc Aug-29-03 3
     RE: Four 3's alexbadmin Sep-07-03 4
         RE: Four 5's Vladimir Sep-13-03 5
         RE: Four 3's Vladimir Sep-15-03 6
             RE: Four 3's alexbadmin Sep-15-03 8
     RE: Four 3's Vladimir Sep-15-03 7
         RE: Four 3's golland Sep-15-03 9

Conferences | Forums | Topics | Previous Topic | Next Topic
Vladimir
Member since Jun-22-03
 Aug-24-03, 08:18 PM (EST)
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1. "RE: Four 3's"
In response to message #0
 
   LAST EDITED ON Aug-24-03 AT 08:26 PM (EST)
 
The only way I can come up with 68 is to cheat with 0.3 and to use the left shift operator << from the programming C-language:

m << n means m·2n

For example:

6 << 3 = 48

The operator << is called left shift, because in binary system it'shifts all 1's of any positive integer m n-times to the left:

6 << 3 = 000001102 << 3 = 001100002 = 48 = 6·23

68 = 3! << 3 + 3!/.3

But once the left and right shift operators are allowed, more possibilities arise to express other integers as four 3's.


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Vladimir
Member since Jun-22-03
 Aug-24-03, 09:41 PM (EST)
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2. "RE: Four 3's"
In response to message #1
 
   I got it without the shift operator:

68 = (3!)!! + (3 + 3)/.3

Was I blind or what?


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sfwc
Member since Jun-19-03
 Aug-29-03, 02:29 PM (EST)
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3. "RE: Four 3's"
In response to message #0
 
   >Cheating with the integer part operator (also called the
>floor function) such as {Ö(3!)} =
>2 or {Ö(33)} = 5 is
>ugly .

Not only is it ugly but it takes out all the fun.
I once proved that any positive integer may be constructed from just 1 three with a suitable nesting of root, factorial and floor functions.

I'm having toruble reconstructing the proof though...
If you have any good ideas to prove this, I would be very grateful.

Thankyou

sfwc
<><


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alexbadmin
Charter Member
1079 posts		 Sep-07-03, 09:00 AM (EST)
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4. "RE: Four 3's"
In response to message #3
 
   >>Cheating with the integer part operator (also called the
>>floor function) such as {Ö(3!)} =
>>2 or {Ö(33)} = 5 is
>>ugly .
>
>Not only is it ugly but it takes out all the fun.
>I once proved that any positive integer may be constructed
>from just 1 three with a suitable nesting of root, factorial
>and floor functions.

Whether this is true or not, finding an actual representation for a given number could be a lot of fun, depending on tastes ...


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Vladimir
Member since Jun-22-03
 Sep-13-03, 11:55 PM (EST)
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5. "RE: Four 5's"
In response to message #4
 
   LAST EDITED ON Sep-22-03 AT 00:29 AM (EST)
 
More four 5's than at https://www.cut-the-knot.org/arithmetic/funny/4_5.shtml

Comments:

1. All expressions have been checked and they are believed to be correct and guaranteed to be different from the current expressions on the above page.

2. I did not erase the checking arithmetics. In case more checking is desired, this arithmetic may be skipviewed before erasing it.

3. I introduced a new perfectly legal operation - a half-root:

0.5Ö(n) = n2

Of course, a half-root is a square, but it can be expressed with the digit 5. For all numbers expressed using the half-root, at least one more expression is given not using the half-root.

4. For clarity of the expressions, I left zeroes in 0.5 = 5/10, in 0.(5) = 0.555555... = 5/9, and in the half-roots. They should be deleted - zero is not five.

5. Originally, I did not intend to use 0.(5). The only number (up to current limit) that seemed to require 0.(5) was 82. Nevertheless, some expressions using 0.(5), including the expression for 82, are groupped at the end in blue color.

6. The current limit is 87, but I did not find any expressions for 71 and 73 below this limit. Twin primes always resist the longest. I will not use the floor function unless required by law.

7. The entire table has been checked again and a number of typos corrected. Mostly formatting, but some arithmetic errors as well.
---------------------------------------------------------------------

1 = 5·5 - 5!/5 = 25 - 120/5 = 25 - 24

2 = 5!! - 5!/5!! - 5 = 15 - 120/15 - 5 = 10 - 8
2 = (5!!/5)!)!!·5/5! = ((15/5)!)!!·5/120 = (3!)!!/24 = 6!!/24 = 48/24
2 = 5!/5!! - (5!!/5)! = 120/15 - (15/5)! = 8 - 3! = 8 - 6
2 = 5!!/5 - 5/5 = 15/5 - 1 = 3 - 1

3 = 5!/5!! - Ö(5·5) = 120/15 - 5 = 8 - 5
3 = 5!!/5 + 5 - 5 = 15/5 + 0
3 = 5.5 - 5·0.5 = 5.5 - 2.5

4 = 5/0.5 - (5!!/5)! = 10 - (15/5)! = 10 - 3! = 10 - 6
4 = 5/5/0.5/0.5 = 1·2·2
4 = 5!!/5 + 5/5 = 15/5 + 1 = 3 + 1

5 = (5 + 5)/0.5 - 5!! = 10·2 - 15
5 = (5!!/5)! - 5/5 = (15/5)! - 1 = 3! - 1 = 6 - 1

6 = 55/5 - 5 = 11 - 5
6 = (5!!/5)! + 5 - 5 = (15/5)! + 0 = 3!

7 = 5!/(5 + 5) - 5 = 120/10 - 5 = 12 - 5
7 = (5!!/5)! + 5/5 = (15/5)! + 1 = 3! + 1 = 6 + 1
7 = 55 - ((5!!/5)!)!! = 55 - ((15/5)!)!! = 55 - (3!)!! = 55 - 6!! = 55 - 48
7 = Ö(5!!·5!!)·0.5 - 0.5 = 15/2 - 0.5 = 7.5 - 0.5
7 = Ö(((5!!/5)!)!/5) - 5 = Ö(6!/5) - 5 = Ö(720/5) - 5 = Ö144 - 5 = 12 - 5

8 = 5!/5!! + 5 - 5 = 120/15
8 = Ö(5!!·5!!)·0.5 + 0.5 = 15/2 + 0.5 = 7.5 + 0.5

9 = 5!! - 5 - 5/5 = 15 - 6
9 = 5!!/5 + 5!!/5 = 15/5 + (15/5) = 3 + 3! = 3 + 6
9 = 5!!/5·5!!/5 = 15/5·15/5 = 3·3
9 = 5!/Ö(5·5) - 5!! = 120/5 - 15 = 24 - 15

10 = 5!!·0.5 + 5·0.5 = 7.5 + 2.5
10 = 5 + 5 + 5 - 5
10 = 55/5.5

11 = 5!! - 5 + 5/5 = 15 - 4
11 = 55/Ö(5·5) = 55/5
11 = ((5!!/5)! - 0.5)/0.5 = ((15/5)! - 0.5)·2 = (3! - 0.5)·2 = (6 - 0.5)·2 = 5.5·2

12 = 5!! - 5!/5!! + 5 = 15 + 5 - 120/15 - 5 = 20 - 8
12 = 5!/Ö(5·5)·0.5 = 120/5/2 = 120/10
12 = Ö(((5!!/5)!)!/Ö(5·5)) = Ö(6!/5) = Ö(720/5) = Ö144

13 = 5!/5!! + Ö(5·5) = 120/15 + 5 = 8 + 5
13 = ((5!!/5)! + 0.5)/0.5 = ((15/5)! + 0.5)·2 = (3! + 0.5)·2 = (6 + 0.5)·2 = 6.5·2

14 = 5!/5 - 5/0.5 = 24 - 10
14 = 5!/5 - 5!! + 5 = 24 - 15 + 5 = 24 - 10
14 = 5!/5!! + (5!!/5)! = 120/15 + (15/5)! = 8 + 3! = 8 + 6

15 = (5!/5!! - 0.5)/0.5 = (120/15 - 0.5) = (8 - 0.5)·2 = 7.5·2
15 = 5!! + 5 - Ö(5·5) = 15 + 5 - 5 = 15 + 0

16 = 55/5 + 5 = 11 + 5
16 = (5!/5!!)!!/5!·5 = (120/15)!!/120·5 = 8!!/24 = 2·4·6·8 / 24 = 2·8
16 = (5!! + 0.5)/0.5 - 5!! = (15 + 0.5)·2 - 15 = 15.5·2 - 15 = 31 - 15

17 = 5!/(5 + 5) + 5 = 120/10 + 5 = 12 + 5
17 = 5·5 - 5!/5!! = 25 - 120/15 - 25 - 8
17 = (5!! - 5 - 0.5)/0.5 = (15 - 5.5)·2 = 8.5·2
17 = Ö(((5!!/5)!)!/5) + 5 = Ö(6!/5) + 5 = Ö(720/5) + 5 = Ö144 + 5 = 12 + 5
17 = (5!/5!! + 0.5)/0.5 = (120/15 + 0.5)·2 = (8 + 0.5)·2 = 8.5·2

18 = 5!! + 5!/5!! - 5 = 15 + 120/15 - 5 = 10 + 8
18 = 5!/5!! + 5 + 5 = 120/15 + 10 = 8 + 10
18 = (5!!/5)·(5!!/5)! = 15/5·(15/5)! = 3·3! = 3·6 
18 = 5!! + (5!!/5)!·0.5 = 15 + (15/5)!/2 = 15 + 3!/2 = 15 + 6/2 = 15 + 3

19 = 5!! + 5 - 5/5 = 15 + 4
19 = (5!! - 5 + 0.5)/0.5 = (15 - 5.5)·2 = 9.5·2

20 = 5!! + Ö(5·5) = 15 + 5
20 = 5!!/0.5 - 5/0.5 = 15·2 - 5·2 = 30 - 10
20 = 5!/5!!·5·0.5 = 120/15·5/2 = 8·2.5 = (5!! + 5·5)·0.5 = (15 + 25)/2 = 40/2

21 = (5!! - 5 + 0.5)/0.5 - 5!! = (15 - 5 + 0.5)·2 = 10.5·2

22 = 55/5/0.5 = 11·2
22 = 5!! + 5!! - 5!/5!! = 15 + 15 - 120/15 = 30 - 8

23 = 5!/5 - 5/5 = 24 - 1
23 = 5!/5!! + 5!! = 120/15 + 15 = 8 + 15
23 = ((5!!/5)!)!! - 5·5 = ((15/5)!)!! - 25 = (3!)!! - 25 = 6!! - 25 = 48 - 25

24 = 5!/5 + 5 - 5 = 24 + 0
24 = 5!/5·5/5 = 24·1
24 = 5!!/0.5 - (5!!/5)! = 15·2 - (15/5)! = 30 - 3! = 30 - 6
24 = (5!! - 0.5)/0.5 - 5 = (15 - 0.5)·2 - 5 = 14.5·2 - 5 = 29 - 5

25 = 5!/5 + 5/5 = 120/5 + 1 = 24 + 1
25 = Ö(5·5·5·5) = 5·5
25 = 5!/5!!·5 - 5!! = 120/15·5 - 15 = 40 - 15
25 = (55 - 5)·0.5 = 50/2
25 = Ö(5! + 5)·Ö5 = Ö125·Ö5 = 5·Ö5·Ö5 = 5·5

26 = 5!! + 55/5 = 15 + 11
26 = (5!/5!! + 5)/0.5 = 120/15 + 5)·2 = (8 + 5)·2 = 13·2
26 = (5!! + 0.5)/0.5 - 5 = (15 + 0.5)·2 - 5 = 15.5·2 - 5 = 31 - 5

27 = 5!! + 5!! - 5!!/5 = 15 + 15 - 15/5 = 30 - 3
27 = 5!! + (5!!/5)!/0.5 = 15 + (15/5)!·2 = 15 + 3!·2 = 15 + 6·2 = 15 + 12
27 = 55·0.5 - 0.5 = 27.5 - 0.5
27 = Ö(((5!!/5)!)!/5) + 5!! = Ö(6!/5) + 15 = Ö(720/5) + 15 = Ö144 + 15 = 12 + 15

28 = 5!! + 5!/5!! + 5 = 15 + 120/15 + 5 = 20 + 8
28 = 55·0.5 + 0.5 = 27.5 + 0.5
28 = (5!! - 5/5)/0.5 = (15 - 1)·2 = 14·2

29 = 5!/5 + Ö(5·5) = 120/5 + 5 = 24 + 5
29 = 5!! + 5!! - 5/5 = 15 + 15 - 1 = 30 - 1

30 = (5 + 5/5)·5 = 6·5
30 = 55 - 5·5 = 55 - 25
30 = (55 + 5)·0.5 = 60/2 = (5.5 + 0.5)·5 = 6·5
30 = 5!/5 + (5!!/5)! = 120/5 + (15/5)! = 24 + 3! = 24 + 6

31 = 5!!/0.5 + 5/5 = 15·2 + 1 = 30 + 1
31 = 55 - 5!/5 = 55 - 120/5 = 55 - 24
31 = 0.5Ö((5!!/5)!) - 5 = (15/5)!)2 - 5 = (3!))2 - 5 = 62 + 5 = 36 - 5

32 = 5!/5!!/0.5/0.5 = 120/15·2·2 = 8·4
32 = 5!/5 + 5!/5!! = 120/5 + 120/15 = 24 + 8
32 = (5!! + 5/5)/0.5 = (15 + 1)·2 = 16·2

33 = 5!! + 5!! + 5!!/5 = 15 + 15 + 15/5 = 30 + 3
33 = 5!/5/0.5 - 5!! = 120/5·2 - 15 = 24·2 - 15 = 48 - 15
33 = 5.5·(5!!/5)! = 5.5·(15/5)! = 5.5·6
33 = (5 + 0.5)· (5!!/5)! = 5.5·(15/5)! = 5.5·6

34 = 5!/5 + 5/0.5 = 120/5 + 10 = 24 + 10
34 = (5!! - 0.5)/0.5 + 5 = (15 - 0.5)·2 + 5 = 14.5·2 + 5 = 29 + 5
34 = 5!/5 + 5!! - 5 = 120/5 + 15 - 5 = 24 + 10

35 = 55 - 5!! - 5 = 50 - 15
35 = 5!/5!!·5 - 5 = 120/15·5 - 5 = 8·5 - 5 = 40 - 5
35 = 5!·0.5 - 5·5 = 120/2 - 25 = 60 - 25
35 = (5!!·0.5 - 0.5)·5 = (15/2 - 0.5)·5 = (7.5 - 0.5)·5 = 7·5

36 = (5!!/5)!·(5!!/5)! = (15/5)!·(15/5)! = (3!)·(3!) = 6·6
36 = 5!!/0.5 + (5!!/5)! = 15·2 + (15/5)! = 30 + 3! = 30 + 6
36 = (5!! + 0.5)/0.5 + 5 = (15 + 0.5)·2 + 5 = 15.5·2 + 5 = 31 + 5

37 = 5!!·5·0.5 - 0.5 = 75/2 - 0.5 = 37.5 - 0.5

38 = ((5!!/5)!)!! - 5 - 5 = ((15/5)!)!! - 10 = (3!)!! - 10 = 6!! - 10 = 48 - 10
38 = ((5!!/5)!)!! - 5!! + 5 = ((15/5)!)!! - 15 + 5 = (3!)!! - 10 = 6!! - 10 = 48 - 10
38 = 5!!·5·0.5 + 0.5 = 75/2 + 0.5 = 37.5 + 0.5

39 = 5!/Ö(5·5) + 5!! = 120/5 + 15 = 24 + 15
39 = (5!! + 5 - 0.5)/0.5 = (15 + 5.5)·2 = 19.5·2

40 = 5!! + 5!! + 5!! - 5 = 3·15 - 5 = 45 - 5
40 = 5!!/0.5 + 5/0.5 = 15·2 + 5·2 = 30 + 10
40 = ((5!!/5)!)!! - 5!/5!! = ((15/5)!)!! - 120/15 = (3!)!! - 8 = 6!! - 8 = 48 - 8
40 = 5/0.5/0.5/0.5 = 5·2·2·2 = 5·8
40 = (5 + 5)/0.5/0.5 =10·2·2 = 10·4
40 = (5!!·0.5 + 0.5)·5 = (15/2 + 0.5)·5 = (7.5 + 0.5)·5 = 8·5

41 = (5!! + 5 + 0.5)/0.5 = (15 + 5.5)·2 = 20.5·2
41 = (5!! + 5.5)/0.5 = (15 + 5.5)·2 = 20.5·2
41 = 0.5Ö((5!!/5)!) + 5 = ((15/5)!)2 + 5 = (3!)2 + 5 = 62 + 5 = 36 + 5

42 = ((5!!/5)!)!! - (5!!/5)! = ((15/5)!)!! - (15/5)! = (3!)!! - 3! = 6!! - 6 = 48 - 6

43 = ((5!!/5)!)!! - Ö(5·5) = ((15/5)!)!! - 5 = (3!)!! - 5 = 6!! - 5 = 48 - 5
43 = 5!/5/0.5 - 5 = 120/5·2 - 5 = 24·2 - 5 = 48 - 5

44 = 5.5·5!/5!! = 5.5·120/15 = 5.5·8
44 = (5 + 0.5)·5!/5!! = 5.5·120/15 = 5.5·8
44 = 5!/5 + 5!! + 5 = 120/5 + 15 + 5 = 24 + 20

45 = 55 - 5 - 5 = 55 - 10
45 = 55 - 5!! + 5 = 55 - 15 + 5 = 40 + 5
45 = 5!/5!!·5 + 5 = 120/15·5 + 5 = 8·5 + 5 = 40 + 5
45 = ((5!!/5)!)!! - 5!!/5 = ((15/5)!)!! - 15/5 = (3!)!! - 3 = 6!! - 3 = 48 - 3
45 = 5·5/0.5 - 5 = 25·2 - 5 = 50 - 5

46 = (5!! + 0.5)/0.5 + 5!! = (15 + 0.5)·2 + 15 = 15.5·2 + 15 = 31 + 15

47 = 55 - 5!/5!! = 55 - 120/15 = 55 - 8
47 = ((5!!/5)!)!! - 5/5 = ((15/5)!)!! - 1 = (3!)!! = 6!! - 1 = 48 - 1
47 = 55 - 5!/5!! = 55 - 120/15 = 55 - 8

48 = ((5!!/5)!)!! + 5 - 5 = ((15/5)!)!! + 0 = (3!)!! = 6!!
48 = (5!!/5)!·5!/5!! = (15/5)!·120/15 = 3!·8 = 6·8
48 = (5!/5!!)!!/(5!/5!!) = (120/15)!!/(120/15)! = 8!!/8 = 2·4·6 = 8·6

49 = ((5!!/5)!)!! + 5/5 = ((15/5)!)!! + 1 = (3!)!! = 6!! + 1 = 48 + 1
49 = 55 - (5!!/5)! = 55 - (15/5)! = 55 - 3! = 55 - 6
49 = 5!/5!! + 5·5 = 120/5 + 25 = 24 + 25
49 = (5·5 - 0.5)/0.5 = (25 - 0.5)·2 = 24.5·2
49 = 0.5Ö(5!/5) - 5!! = (120/15)2 - 15 = 82 - 15 = 64 - 15

50 = 55 - Ö(5·5) = 55 - 5
50 = 5!!·5 - 5·5 = 15·5 - 25 = 75 - 25
50 = 5!! + 5!! + 5!! + 5 = 3·15 + 5 = 45 + 5
50 = (5! - 5!! - 5)·0.5 = (120 - 15 - 5)/2 = 100/2

51 = ((5!!/5)!)!! + 5!!/5 = ((15/5)!)!! + 15/5 = (3!)!! + 3 = 6!! + 3 = 48 + 3
51 = 5!!·5 - 5!/5 = 15·5 - 120/5 = 75 - 24
51 = (5·5 + 0.5)/0.5 = (25 + 0.5)·2 = 25.5·2

52 = 55 - 5!!/5 = 55 - 15/5 = 55 - 3
52 = 5!·0.5 - 5!/5!! = 120/2 - 120/15 = 60 - 8
52 = (5! - 5!!)·0.5 - 0.5 = (120 - 15)/2 - 0.5 = 105/2 - 0.5 = 52.5 - 0.5

53 = ((5!!/5)!)!! + Ö(5·5) = ((15/5)!)!! + 5 = (3!)!! + 5 = 6!! + 5 = 48 + 5
53 = 5!/5/0.5 + 5 = 120/5·2 + 5 = 24·2 + 5 = 48 + 5
53 = (5! - 5!!)·0.5 + 0.5 = (120 - 15)/2 + 0.5 = 105/2 + 0.5 = 52.5 + 0.5

54 = 55 - 5/5
54 = ((5!!/5)!)!! + (5!!/5)! = ((15/5)!)!! + (15/5)! = (3!)!! + 3! = 6!! + 6 = 48 + 6
54 = 5!·0.5 - (5!!/5)! = 120/2 - (15/5)! = 60 - 3! = 60 - 6
54 = 5!/5 + 5!! + 5!! = 120/5 + 15 + 15 = 24 + 30

55 = 55 + 5 - 5
55 = 5!·0.5 - Ö(5·5) = 120/2 - 5 = 60 - 5
55 = (5!! + 5!!)/0.5 - 5 = (15 + 15)·2 + 5 = 60 - 5
55 = 5!!·5 - 15!! - 5 = 15·5 - 20 = 75 - 20
55 = (5! - 5!! + 5)·0.5 = (120 - 15 + 5) = 110/2
55 = 5·5/0.5 + 5 = 25·2 + 5 = 50 + 5

56 = 55 + 5/5 = 55 + 1
56 = ((5!!/5)!)!! + 5!/5!! = ((15/5)!)!! + 120/15 = (3!)!! + 8 = 6!! + 8 = 48 + 8

57 = 5!·0.5 - 5!!/5 = 120/2 - 15/5 = 60 - 3
57 = (5! - 5)·0.5 - 0.5 = (120 - 5)/2 - 0.5 = 115/2 - 0.5 = 57.5 - 0.5
57 = 5! - ((5!!/5)!)!! - 5!! = 120 - ((15/5)!)!! - 15 = 105 - (3!)!! = 105 - 6!! = 105 - 48

58 = 55 + 5!!/5 = 55 + 15/5 = 55 + 3
58 = (5! - 5)·0.5 + 0.5 = (120 - 5)/2 + 0.5 = 115/2 + 0.5 = 57.5 + 0.5
58 = ((5!!/5)!)!! + 5 + 5 = ((15/5)!)!! + 10 = (3!)!! + 10 = 6!! + 10 = 48 + 10

59 = 5!·0.5 - 5/5 = 120/2 - 1 = 60 - 1
59 = (5!! + 5!! - 0.5)/0.5 = (2·15 - 0.5)·2 = (30 - 0.5)·2 = 29.5·2
59 = 0.5Ö(5!/5) - 5 = (120/15)2 - 5 = 82 - 5 = 64 - 5

60 = 55 + Ö(5·5) = 55 + 5
60 = 5!!/0.5 + 5!!/0.5 = 15·2 + 15·2 = 30 + 30
60 = Ö(((5!!/5)!)!/5)·5 = Ö(6!/5)·5 = Ö(720/5)·5 = Ö144·5 = 12·5
60 = Ö(5·5!!·((5!!/5)!)!!) = Ö(5·15·((15/5)!)!!) = Ö(5·15·(3!)!!) = Ö(5·15·6!!) = Ö(5·15·48) = Ö(25·9·16) = 5·3·4 = 15·4

61 = 5!·0.5 + 5/5 = 120/2 + 1 = 60 + 1
61 = 55 + (5!!/5)! = 55 + (15/5)! = 55 + 3! = 55 + 6
61 = (5!! + 5!! + 0.5)/0.5 = (2·15 + 0.5)·2 = (30 + 0.5)·2 = 30.5·2

62 = (5! + 5)·0.5 - 0.5 = 120 + 5)/2 - 0.5 = 125/2 - 0.5 = 62.5 - 0.5

63 = 55 + 5!/5!! = 55 + 120/15 = 55 + 8
63 = 5!/5/0.5 + 5!! = 120/5·2 + 15 = 24·2 + 15 = 48 + 15
63 = 5!·0.5 + 5!!/5 = 120/2 + 15/5 = 60 + 3
63 = (5! + 5)·0.5 + 0.5 = (120 + 5)/2 + 0.5 = 125/2 + 0.5 = 62.5 + 0.5

64 = 5!/5!!·5!/5!! = 120/15·120/15 = 8·8
64 = (5!/5!!)!!/(5!!/5)! = (120/15)!!/(15/5)! = 8!!/3! = 2·4·6·8 / 6 = 8·8

65 = 5!·0.5 + Ö(5·5) = 120/2 + 5 = 60 + 5
65 = (5!! + 5!!)/0.5 + 5 = (15 + 15)·2 + 5 = 60 + 5
65 = 5!!/0.5/0.5 + 5 = 15·2·2 + 5 = 60 + 5 = (5! + 5!! - 5)·0.5 = (120 + 15 - 5)/2 = 130/2

66 = 5!·0.5 + (5!!/5)! = 120/2 + (15/5)! = 60 + 3! = 60 + 6
66 = (((5!!/5)!)!! - 5!!)/0.5 = (((15/5)!)!! - 15)·2 = ((3!)!! - 15)·2 = (6!! - 15)·2 = (48 - 15)·2 = 33·2

67 = 5!!·5 - 5!/5!! = 15·5 - 120/15 = 75 - 8
67 = (5! + 5!!)·0.5 - 0.5 = (120 + 15)/2 - 0.5 = 135/2 - 0.5 = 67.5 - 0.5
67 = 5! - ((5!!/5)!)!! - 5 = 120 - ((15/5)!)!! - 5 = 115 - (3!)!! = 115 - 6!! = 115 - 48

68 = 5!·0.5 + 5!/5!! = 120/2 + 120/15 = 60 + 8
68 = (5! + 5!!)·0.5 + 0.5 = (120 + 15)/2 + 0.5 = 135/2 + 0.5 = 67.5 + 0.5

69 = 5!!·5 - (5!!/5)! = 15·5 - (15/5)! = 75 - 3! = 75 - 6
69 = 0.5Ö(5!/5) + 5 = (120/15)2 + 5 = 82 + 5 = 64 + 5

70 = 5!!·Ö(5·5) - 5 = 15·5 - 5 = 75 - 5
70 = (5! + 5!! + 5)·0.5 = (120 + 15 + 5)/2 = 140/2

71 = ???

72 = ((5!!/5)!)!! + 5!/5 = ((15/5)!)!! + 120/5 = (3!)!! + 24 = 6!! + 24 = 48 + 24
72 = 5!!·5 - 5!!/5 = 15·5 - 15/5 = 75 - 3
72 = 5!!/5·5!/5 = 15/5·120/5 = 3·24
72 = ((5!!/5)!)!/5·0.5 = ((15/5)!)!/5/2 = (3!)!/10 = 6!/10 = 720/10
 
73 = ???

74 = 5!!·5 - 5/5 = 15·5 - 1 = 75 - 1

75 = 55 + 5!! + 5 = 60 + 15
75 = 5!!·5 + 5 - 5 = 15·5 + 0
75 = 5!!·(5!!/5)! - 5!! = 15·(15/5)! - 15 = 15·3! - 15 = 15·6 - 15 = 90 - 15

76 = 5!!·5 + 5/5 = 15·5 + 1 = 75 + 1

77 = 5! - ((5!!/5)!)!! + 5 = 120 - ((15/5)!)!! + 5 = 125 - (3!)!! = 125 - 6!! = 125 - 48

78 = ((5!!/5)!)!! + 5!!/0.5 = ((15/5)!)!! + 15·2 = (3!)!! + 30 = 6!! + 30 = 48 + 30
78 = 5!!·5 + 5!!/5 = 15·5 + 15/5 = 75 + 3

79 = 55 + 5!/5 = 55 + 24
79 = 0.5Ö(5!/5) + 5!! = (120/15)2 + 15 = 82 + 15 = 64 + 15

80 = 55 + 5·5 = 55 + 25
80 = 5!!·5 + Ö(5·5) = 15·5 + 5 = 75 + 5
80 = (55 - 5!!)/0.5 = (55 - 15)·2 = 40·2
80 = (5·5 - 5!!)/0.5 = (25 + 15)·2 = 40·2
80 = 5! - 5·5 - 5!! = 120 - 25 - 15 = 120 - 40

81 = 5!!·5 + (5!!/5)! = 15·5 + (15/5)! = 75 + 3! = 75 + 6
81 = ((5!!/5)!)!!/0.5 - 5!! = ((15/5)!)!!·2 - 15 = (3!)!!·2 - 15 = 6!!·2 - 15 = 48·2 - 15 = 96 - 15

82 - see 0.(5) = 5/9 cases

83 = 5!!·5 + 5!/5!! = 15·5 + 120/15 = 75 + 8

84 = 5! - 0.5Ö((5!!/5)!) = 120 - ((15/5)!)2 = 120 - (3!)2 = 120 - 62 = 120 - 36

85 = 5!!·5 + 5 + 5 = 15·5 + 10 = 75 + 10
85 = 5!!·(5!!/5)! - 5 = 15·(15/5)! - 5 = 15·3! - 5 = 15·6 - 5 = 90 - 5

86 = (((5!!/5)!)!! - 5)/0.5 = (((15/5)!)!! - 5)·2 = ((3!)!! - 5)·2 = (6!! - 5)·2 = (48 - 5)·2 = 43·2

87 = 5! - ((5!!/5)!)!! + 5!! = 120 - ((15/5)!)!! + 15 = 135 - (3!)!! = 135 - 6!! = 135 - 48

---------------------------------------------------------------------

The following expressions use 0.(5) = 0.555555... = 5/9.

 
17 = 5!!/0.(5) - 5 - 5 = 15·9/5 - 10 = 27 - 10
17 = 5!!/0.(5) - 5!! + 5 = 15·9/5 - 15 + 5 = 27 - 10
17 = (5!!/)0.(5) - 0.5)/0.5 = (15·9/5 - 0.5)·2 = 8.5·2

18 = 5/0.(5) + 5/0.(5) = 5·9/5 + 5·9/5 = 9 + 9

19 = (5!!/)0.(5) + 0.5)/0.5 = (15·9/5 + 0.5)·2 = 9.5·2

21 = ((5!!/5)!)!! - 5!!/0.(5) = (15/5)!)!! - 5·9/5 = (3!)!! - 27 = 6!! - 27 = 48 - 27

28 = 55 - 5!!/0.(5) = 55 - 15·9/5 = 55 - 27

37 = 5!!/0.(5) + 5!! - 5 = 15·9/5 + 15 - 5 = 27 + 10

46 = 55 - 5/0.(5) = 55 - 5·9/5 = 55 - 9

47 = 5!!/0.(5) + 5!! + 5 = 15·9/5 + 15 + 5 = 27 + 20

50 = (5!!·5 + 5!!)·0.(5) = (15·5 + 15)·5/9 = 6·15·5/9 = 2·5·5 = 2·25

51 = 5!/5 + 5!!/0.(5) = 120/5 + 15·9/5 = 24 + 27

53 = (5!!/0.(5) - 0.5)/0.5 = (15·9/5 - 0.5)·2 = (27 - 0.5)·2 = 26.5·2

54 = 5!!/0.(5) + 5!!/0.(5) = 15·9/5 + 15·9/5 = 27 + 27

55 = (5!!/0.(5) + 0.5)/0.5 = (15·9/5 + 0.5)·2 = (27 + 0.5)·2 = 27.5·2

57 = 5!!/0.(5) + 5!! + 5!! = 15·9/5 + 15 + 15 = 27 + 30

64 = 55 + 5/0.(5) = 55 + 5·9/5 = 55 + 9

66 = 5!!/5 - 5/0.(5) = 15·5 - 5·9/5 = 75 - 9
66 = 0.5Ö(5/0.(5)) - 5!! = (5·9/5)2 - 15 = 92 - 15 = 81 - 15

75 = ((5!!/5)!)!! + 5!!/0.(5) = ((15/5)!)!! + 5·9/5 = (3!)!! + 27 = 6!! + 27 = 48 + 27

76 = 0.5Ö(5/0.(5)) - 5 = (5·9/5)2 - 5 = 92 - 5 = 81 - 5

78 = 5! - 5!!/0.(5) - 5!! = 120 - 15·9/5 - 15 = 120 - 27 - 15 = 120 - 42

80 = ((5!!/5)!)!/5·0.(5) = ((15/5)!)!/5·5/9 = (3!)!/9 = 6!/9 = 720/9

81 = 5·5/0.(5)/0.(5) = 25·9/5·9/5 = 9·9

82 = 55 + 5!!/0.(5) = 55 + 15·9/5 = 55 + 27

86 = 0.5Ö(5/0.(5)) + 5 = (5·9/5)2 + 5 = 92 + 5 = 81 + 5

Look Mom, no floor functions!


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Vladimir
Member since Jun-22-03
 Sep-15-03, 07:47 AM (EST)
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6. "RE: Four 3's"
In response to message #4
 
   LAST EDITED ON Sep-16-03 AT 00:41 AM (EST)
 
I was reading the page One 4 and I have a few questions (no hurry):

7 = g4 - the 4-th gnomic number

The definition of Gnomic Numbers immediately leads to their equivalence with odd numbers. But then you say "But the fourth odd number is 7, not 5. So I am not sure which gnomic number Robert Smith had in mind." Is not 7 = g4 (and not 5 = g4) in his table?

BTW, number 7 in the One 4 table could also be the 4-th Lucas Number L4 = 7. Lucas numbers are defined by the same recurrence formula as Fibonacci numbers:

Ln+1 = Ln + Ln-1

but with a different seed (L1 = 1, L2 = 3) as opposed to the Fibonacci sequence seed (F1 = 1, F2 = 1).

10 = T4 - the 4-th tetrahedral number

This does not seem to be correct. Thetrahedral Numbers are usually denoted Ten (not Tn) and defined as

Ten = Snk=1 Tk

where Tk = k(k + 1)/2 are Triangular Numbers. Moreover, the 4-th tetrahedral number is Te4 = 20 (1, 4, 10, 20, 35, ...), while the 4-th triangular number is T4 = 10 (1, 3, 6, 10, 16, ...), the correct value for number 10 in the One 4 table.


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alexbadmin
Charter Member
1079 posts		 Sep-15-03, 08:40 PM (EST)
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8. "RE: Four 3's"
In response to message #6
 
   LAST EDITED ON Sep-15-03 AT 08:42 PM (EST)
 
>The definition of
>Gnomic
>Numbers immediately leads to their equivalence with odd
>numbers. But then you say "But the fourth odd number is 7,
>not 5. So I am not sure which gnomic number Robert Smith had
>in mind." Is not 7 = g4 (and not 5 =
>g4) in his table?

It is indeed.

>BTW, numer 7 in the One 4 table could also be the 4-th
>Lucas Number

...

>L4 = 7. >10 = T4 - the 4-th tetrahedral number
>
>This does not seem to be correct.

I made a correction and a footnote on the page. Thank you.


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Vladimir
Member since Jun-22-03
 Sep-15-03, 07:47 AM (EST)
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7. "RE: Four 3's"
In response to message #3
 
   The only number increasing increasing operation is the factorial. Consider a huge number obtained by applying the factorial on any n < m as may times as we wish (n < m with an induction part of the proof in mind)

N = (...(n)!)!...)!

The the only decreasing operation is the square root and the floor function to stay within integers:

x = N-2k
m = floor(x)

Taking a double logarithm with any base (say 2) of x:

log2(log2(x)) = log2(log2(N-2k)) =
log2(-2k·log2(N)) = log2(log2(N)) - k

it appears that we need log2(log2(N)) - k to be dense in the interval (1, 2), where N = (...(n)!)!...)!, n is any integer n < m, and k is any integer. This is equivalent to k - log2(log2(N)) being dense in the interval (0, 1). For that, we need the mantisa of 1/N to be dense in (0, 1). As we perform the factorial on n < m as many times as we wish, N devours more and more primes, and all possible periods in the mantisa of 1/N should show up.

These are just vague thoughts indicating that I do not have the math tools to prove the conjecture - as a physicist, I know virtually no number theory.

Sorry, Vladimir


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golland
guest
 Sep-15-03, 10:44 PM (EST)
 
9. "RE: Four 3's"
In response to message #7
 
   Hello Vladimir,

I have not noticed a use of subfactorial (the number of derangements)
It could be helpful in creation of some numbers.

!1=0, !2=1, !3=2, !4=9, !5=44, !6=265

Good luck.

G.


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