March 31st, 2017, 03:15 AM  #1 
Member Joined: Aug 2015 From: Chiddingfold, Surrey Posts: 42 Thanks: 1 Math Focus: Number theory, Applied maths  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 â€“ INT ((p )/3) + 1 27 â€“ 54 2* (p â€“ INT ((p + 1.5)/3)  1 53 â€“ 80 2* (p â€“ INT ((p  0.5)/3) â€“ 1 79  109 2* (p â€“ INT ((p + 0.5)/3)  1 109  ??? 2* (p â€“ INT ((p + 1.5)/3) â€“ 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. 
March 31st, 2017, 09:00 PM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,353 Thanks: 464 Math Focus: Yet to find out. 
It's hard to read. If you typeset your equations with LaTeX it would be better.


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