Leibnitz Prime Function
Define the function N’ on the integers by the Leibnitz rule:
(ab)’ = a’b + ab’
Exercise for the student: Prove from this:
(a^n)’ = n (a^(n-1)) a’
0’ = 1’ = (-1)’ = 0
(-a)’ = -(a’)
To complete the definition, assume also:
(Prime)’ = 1
This is not linear: (2+3)’=5’=1 ; but 2’+3’=1+1= 2
N N’
1 = 0
2 = 1
3 = 1
4 2*(2^1)*2’ = 4
5 = 1
6 2’3+2*3’ = 5
7 = 1
8 3*(2^2)*2’ = 12
9 2*(3^1)*3’ = 6
10 2’5+5’2 = 7
11 = 1
12 (2^2)’3+(2^2)*3’ = 16
13 = 1
14 2’7 + 7’2 = 9
15 3’5 + 5’3 = 8
16 4*(2^3)*2’ = 32
17 = 1
18 (3^2)’*2 + (3^2)*2’ = 21
19 = 1
20 (2^2)’*5 + (2^2)*5’ = 24
21 3’7 + 7’3 = 10
22 2’11+11’2 = 13
23 = 1
24 8’3+3’8 = 36+8 = 44
25 2*5*5’ = 10
26 2’13+13’2 = 15
27 3*(3^2)*3’ = 27
28 4’7+7’4 = 28+4 = 32
29 = 1
30 2’*3*5+2*3’*5+2*3*5’ = 31
31 = 1
32 5*2^4*2’ = 80
33 3’11+11’3 = 14
34 2’17+17’2 = 19
35 5’7 + 7’5 = 12
36 4’9+9’4 = 36+24 = 60
37 = 1
38 2’19+19’2 = 21
39 3’13 + 13’3 = 16
40 8’5+5’8 = 60+8 = 68
Define N’ on the rational numbers by:
(a/b)’ = (a’b – b’a)/(b^2)
It is well-defined with respect to rescaling:
((ka)/(kb))’ = ( (ka)’kb – (kb)’ka) ) / (kb)^2
= ( (k’a+a’k)kb – (k’b+b’k)ka) ) / (kb)^2
= ( (k’kba+a’kkb – k’bka - b’kka) ) / (kb)^2
= ( a’kkb - b’kka) ) / ((k^2)(b^2)
= ( a’b - b’a) ) / (b^2)
= (a/b)’
(4/3)’ = ( 4’3-3’4) / 3^2 = (4*3-4)/9 = 8/9
(35/6)’ = (35’6 – 6’35) / (6^2)
= (12*6 – 5*35) / 36 = - 103/36
(27/4)’ = (27’4 - 4’27)/(4^2)
= (27*4 - 4*27)/16 = 0
Exercises for the student: Prove:
Theorem: if 4 divides into N then N’ > N
Theorem: if P is prime then (P^P)’ = (P^P)
Theorem: If N’=N and n’=n then (n/N)’ = 0
Theorem: If Q’=0 and R’=0 then (PQ)’=(P/Q)’= 0
Theorem: If R’=0 then for any N, (PN)’=P(N’)
Speculation: Between any P and Q there is an R such that R’ = 0.
Speculation: if N is odd then N’ < N
Question: for what N does the sequence N, N’, N’’, N’’’, …
… diverge to infinity? Reach 0 or other fixed value? Cycle?
Question: Given N, can you solve M’ = N?
Question: What is the Leibnitz prime function for?
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