Consider all possible pairs (M, N) of positive integers that satisfy this equation:
M2 = N(M+N) - 1
• If this equation has a finite number of solutions, then list all of them.
• If the total number of solutions is infinite, then find the general forms of M and N with valid reasoning.
Note: Extra credit wii be given for a semi-analytic (hand calculator and p&p) solution.
m^2 = n(m+n) - 1
m^2 - n*m - n^2 + 1 = 0
quadratic for m given constant n.
For n from 1 to 1,000,000:
for n=1:1000000 % given n, solving for m
s=roots([1,-n,1-n^2]); % polynomial coefficients
s1=sprintf('%14.14f',s(1));
s2=sprintf('%14.14f',s(2));
s=[str2double(s1) str2double(s2)];
if s==round(s)
disp([s n])
end
end
Possible M values N
0 1 1
3 -1 2
8 -3 5
21 -8 13
55 -21 34
144 -55 89
377 -144 233
987 -377 610
6765 -2584 4181
17711 -6765 10946
46368 -17711 28657
317811 -121393 196418
832040 -317811 514229
Values of (M,N) are (F(2*i),F(2*i-1)) where F(n) is the nth Fibonacci number, named after Leonardo of Pisa. Note that the first row has its positive M in the second column rather than the first.
The pattern does continue, with non-positive values, when N is non-positive:
-144 55 -89
-55 21 -34
-21 8 -13
-8 3 -5
-3 1 -2
0 -1 -1
Note: the solution for N = 0 has an imaginary M = +/- i.
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Posted by Charlie
on 2023-02-21 15:05:55 |
computer solution
