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March 31st, 2017, 04:15 AM   #1
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From: Chiddingfold, Surrey

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Math Focus: Number theory, Applied maths
Question An extension to FLT

A THEORETICAL EXTENSION TO FLT
1. The smallest sum of a set of four different positive integers A B C D that are related by A = INT(( Bp + Cp )1/p +0.5) and D = INT(( Ap - Bp - Cp )1/p + 0.5), A B and C being relatively prime, IS WHEN B - C = 1
2. The value of A is an expression in any positive integer p.
3. The expressions I have established by trial for p from 1 to 113 are as follows:
P expression
1 - 8 2* p + 1
8 - 28 2* (p – INT ((p )/3) + 1
27 – 54 2* (p – INT ((p + 1.5)/3) - 1
53 – 80 2* (p – INT ((p - 0.5)/3) – 1
79 - 109 2* (p – INT ((p + 0.5)/3) - 1
109 - ??? 2* (p – INT ((p + 1.5)/3) – 1

Observations:
The calculated value of A is equal to B + 1
The smallest value of D is where the fractional part of A is nearest and above 0.5 except when p = 1 or 2 when D = 0
Expressions for values of p higher than 113 cannot be determined without a computer having a higher resolution than the usual 64 bit.
The division by 3 will presumably apply to all values of p and is due to the fact that for four consecutive values of p above 8, there are only three different values of A one of which is duplicated.
Example
p A
18 25
19 27
20 29
21 29


There is an overlap between adjacent expressions since both expressions produce the same result. This is where two identical values of A are followed by another two. Elsewhere there is one other value between them.
Example
p A
26 37
27 37
28 39
29 39

I'd like to find a proof of my theory and would like to know if anyone can find a counter example.
magicterry is offline  
 
March 31st, 2017, 10:00 PM   #2
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It's hard to read. If you typeset your equations with LaTeX it would be better.
Joppy is offline  
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