Five positive integers A, B, C, D and E
, with A < B < C < D < E
, are such that:
(i) A, B and C
(in this order) are in arithmetic sequence, and:
(ii) B, C and D
(in his order) are in geometric sequence, and:
(iii) C, D and E
(in this order) are in harmonic sequence
Determine the minimum value
such that there are precisely two quintuplets (A, B, C, D, E)
that satisfy all the given conditions.
: Try to solve this problem analytically, although computer program/ spreadsheet solutions are welcome.
(In reply to computer solutions -- no proof
Using substitution I was able to derive the equation,
[E-A] = (4/A)*x2 + 4*x,
such that x is the common difference between A and B.
With this bit of information, and limiting my search to where
[E-A] = 64, I was able to search beyond A=200,000 to find if there were any subsequent sets. Yet, due to limitations, my search limited A to not much greater than 2,000,000, i.e., only a 10-fold increase in range. No values for (A, x) other than (8, 8) and (36, 12) such that D also was an integer, i.e.,
x2 modulo (A + x) = 0, was found.
Posted by Dej Mar
on 2008-10-29 10:09:47