Determine all possible integer values of N that satisfy this equation:
(√5 + 2)
N + (√5 - 2)
N= 18
clc
digits 50
a=sym(sqrt(sym(5))+2);
b=sym(sqrt(sym(5))-2);
for n=1:30
v=a^n+b^n;
v2=vpa(a^n+b^n);
fprintf('%3d %s %s\n',n,v2,v)
end
N = 2 as shown below:
N LHS (numeric and symbolic)
1 4.4721359549995793928183473374625524708812367192231 2*5^(1/2)
2 18.0 (5^(1/2) - 2)^2 + (5^(1/2) + 2)^2
3 76.026311234992849677911904736863392004981024226792 (5^(1/2) - 2)^3 + (5^(1/2) + 2)^3
4 322.0 (5^(1/2) - 2)^4 + (5^(1/2) + 2)^4
5 1364.001466274871714809595937926078503618777199363 (5^(1/2) - 2)^5 + (5^(1/2) + 2)^5
6 5778.0 (5^(1/2) - 2)^6 + (5^(1/2) + 2)^6
7 24476.000081712698016894814977932549673133008564308 (5^(1/2) - 2)^7 + (5^(1/2) + 2)^7
8 103682.0 (5^(1/2) - 2)^8 + (5^(1/2) + 2)^8
9 439204.00000455369258929707366485981561277537695818 (5^(1/2) - 2)^9 + (5^(1/2) + 2)^9
10 1860498.0 (5^(1/2) - 2)^10 + (5^(1/2) + 2)^10
11 7881196.0000002537685904525109895441313568237766829 (5^(1/2) - 2)^11 + (5^(1/2) + 2)^11
12 33385282.0 (5^(1/2) - 2)^12 + (5^(1/2) + 2)^12
13 141422324.00000001414203884812414693454881005260333 (5^(1/2) - 2)^13 + (5^(1/2) + 2)^13
14 599074578.0 (5^(1/2) - 2)^14 + (5^(1/2) + 2)^14
15 2537720636.0000000007881088137236552777472241230833 (5^(1/2) - 2)^15 + (5^(1/2) + 2)^15
16 10749957122.0 (5^(1/2) - 2)^16 + (5^(1/2) + 2)^16
17 45537549124.000000000043919798901648064901224162896 (5^(1/2) - 2)^17 + (5^(1/2) + 2)^17
18 192900153618.0 (5^(1/2) - 2)^18 + (5^(1/2) + 2)^18
19 817138163596.00000000000244756650600989047481080905 (5^(1/2) - 2)^19 + (5^(1/2) + 2)^19
20 3461452808002.0 (5^(1/2) - 2)^20 + (5^(1/2) + 2)^20
21 14662949395604.0000000000001363982065299636453704 (5^(1/2) - 2)^21 + (5^(1/2) + 2)^21
22 62113250390418.0 (5^(1/2) - 2)^22 + (5^(1/2) + 2)^22
23 263115950957276.00000000000000760121152945514185639 (5^(1/2) - 2)^23 + (5^(1/2) + 2)^23
24 1114577054219522.0 (5^(1/2) - 2)^24 + (5^(1/2) + 2)^24
25 4721424167835364.0000000000000004236010002289080447 (5^(1/2) - 2)^25 + (5^(1/2) + 2)^25
26 20000273725560978.0 (5^(1/2) - 2)^26 + (5^(1/2) + 2)^26
27 84722519070079276.000000000000000023606474665202947 (5^(1/2) - 2)^27 + (5^(1/2) + 2)^27
28 358890350005878082.0 (5^(1/2) - 2)^28 + (5^(1/2) + 2)^28
29 1520283919093591604.000000000000000001315543744745 (5^(1/2) - 2)^29 + (5^(1/2) + 2)^29
30 6440026026380244498.0 (5^(1/2) - 2)^30 + (5^(1/2) + 2)^30
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Posted by Charlie
on 2023-10-02 09:24:48 |
solution
