$a^{12}$ + $b^{12}$ + $c^{12}$ + $d^{12}$ + $e^{12}$ wanted

Let $P_n = a^n + b^n + c^n + d^n + e^n$ , where $n$ is a positive integer, and

$S_1 = \displaystyle \sum_{cyc}^5 a = P_1 = 1 \quad \quad S_2 = \sum_{cyc}^5 ab \quad \quad S_3 = \sum_{cyc}^5 abc \quad \quad S_4 = \sum_{cyc}^5 abcd \quad \quad S_5 = abcde$

Using Newton sums method, we have:

\(\begin{array} {} P_1 = S_1 = 1 & \Rightarrow S_1 = 0 \\ P_2 = S_1P_1 - 2S_2 = 1 & \Rightarrow S_2 = 0 \\ P_3 = S_1P_2 - S_2P_1 + 3S_3 = 2 & \Rightarrow S_3 = \frac{1}{3} \\ P_4 = S_1P_3 - S_2P_2 + S_3P_1 - 4S_4 = 3 & \Rightarrow S_4 = - \frac{1}{6} \\ P_5 = S_1P_4 - S_2P_3 + S_3P_2 - S_4P_1 + 5S_5 = 5 & \Rightarrow S_5 = \frac{3}{10} \end{array} \)

For $n > 5$ , we have $P_n = S_1P_{n-1} - S_2P_{n-2} + S_3P_{n-3} - S_4P_{n-4} + S_5P_{n-5}$ .

Using the following Excel spreadsheet we found that $P_{12} = a^{12} + b^{12} + c^{12} + d^{12} + e^{12} = \dfrac{47147}{1350}$ .

Therefore, $m-n = \boxed{45797}$

S1 = 1; 1: P1 = 1

S2 = 0; 1: P2 = 1

S3 = 1/3; 2: P3 = 2

S4 = -1/6; 3: P4 = 3

S5 = 0.3; 5: P5 = 5

S6 = 0; 8: P6 = 6.13333333333333 92 15 77

S7 = 0; 13: P7 = 7.76666666666667 233 30 203

S8 = 0; 21: P8 = 10.5333333333333 158 15 143

S9 = 0; 34: P9 = 14.3111111111111 644 45 599

S10 = 0; 55: P10 = 19.4222222222222 874 45 829

S11 = 0; 89: P11 = 26.0677777777778 23461 900 22561

S12 = 0; 144: P12 = 34.9237037037037 47147 1350 45797

363127

The non-mentioned things behind these are actually Complex Numbers which allow all these to happen in validity; once a, b, c, d and e are fixed, all are just fixed:

a = 1.344665235633880

b = 0.276584727888310 + j 0.699496514084196

c = 0.276584727888310 - j 0.699496514084196

d = -0.448917345705252 + j 0.439081791607579

e = -0.448917345705252 - j 0.439081791607579

However, they are not the preference for evaluation of a^n + b^n + c^n + d^n + e^n but Newton sums method is the way; evaluation using Excel can be most accurate for n of high power. Nevertheless, COMPLEX and IMPOWER and etc allow us to do arithmetic of complex numbers in Microsoft Excel.