# Does this trigonometric pattern continue for all primes $p=6m+1$?

(Revised from its original form.) Consider primes $p=6m+1$, and $p>13$:

$p=19=3\times\color{blue}{6}+1$:

Let $\beta = 2\pi/19.\,$ A root of $x^3+x^2-\color{blue}{6}x-7=0$ is $$x=2\big(\cos(2\beta)+\cos(3\beta)+\cos(5\beta)\big)$$
Note that $2+3=5$, or the last multiplier is the sum of the first two.

$p=31=3\times\color{blue}{10}+1$

Let $\beta = 2\pi/31.\,$ A root of $x^3+x^2-\color{blue}{10}x-8=0$ is $$x=2\big(\cos(2\beta)+\cos(4\beta)+\cos(8\beta)+\cos(16\beta)+\cos(30\beta)\big)$$
Similarly, $2+4+8+16=30$. Note that $\cos(30\beta) =\cos(32\beta)$.

$p=37=3\times\color{blue}{12}+1$

Let $\beta = 2\pi/37.\,$ A root of $x^3+x^2-\color{blue}{12}x+11=0$ is $$x=2\big(\cos(2\beta)+\cos(9\beta)+\cos(12\beta)+\cos(15\beta)+\cos(16\beta)+\cos(54\beta)\big)$$
Again, $2+9+12+15+16=54$.

$p=43=3\times\color{blue}{14}+1$

Let $\beta = 2\pi/43.\,$ A root of $x^3+x^2-\color{blue}{14}x+8=0$ is $$x=2\sum_{k=1}^7\cos\big(2^k\beta)$$
But alternatively,
$$x=2\big(\cos(\beta)+\cos(4\beta)+\cos(11\beta)+\cos(16\beta)+\cos(21\beta)+\cos(35\beta)+\cos(88\beta)\big)$$
and, $1+4+11+16+21+35=88$.

Question: Is it true that argument multipliers of the cubic roots obey $\displaystyle\sum_{k=1}^{m-1}a_k=a_m$ for any $p=6m+1$?

#### Solutions Collecting From Web of "Does this trigonometric pattern continue for all primes $p=6m+1$?"

made a loop to perform the method of Gauss; this is the short version, primes $p \equiv 1 \pmod 3$ up to $1000$

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jagy@phobeusjunior:~$./cubic_cyclic_gauss_loop | grep exps x^3 + x^2 - 2 x - 1 p 7 p.root 3 exps 6^k d = 7^2 x^3 + x^2 - 4 x + 1 p 13 p.root 2 exps 5^k d = 13^2 x^3 + x^2 - 6 x - 7 p 19 p.root 2 exps 8^k d = 19^2 x^3 + x^2 - 10 x - 8 p 31 p.root 3 exps 15^k d = 2^2 * 31^2 x^3 + x^2 - 12 x + 11 p 37 p.root 2 exps 8^k d = 37^2 x^3 + x^2 - 14 x + 8 p 43 p.root 3 exps 2^k d = 2^2 * 43^2 x^3 + x^2 - 20 x - 9 p 61 p.root 2 exps 8^k d = 3^2 * 61^2 x^3 + x^2 - 22 x + 5 p 67 p.root 2 exps 3^k d = 3^2 * 67^2 x^3 + x^2 - 24 x - 27 p 73 p.root 5 exps 7^k d = 3^2 * 73^2 x^3 + x^2 - 26 x + 41 p 79 p.root 3 exps 12^k d = 79^2 x^3 + x^2 - 32 x - 79 p 97 p.root 5 exps 19^k d = 97^2 x^3 + x^2 - 34 x - 61 p 103 p.root 5 exps 3^k d = 3^2 * 103^2 x^3 + x^2 - 36 x - 4 p 109 p.root 6 exps 2^k d = 2^4 * 109^2 x^3 + x^2 - 42 x + 80 p 127 p.root 3 exps 5^k d = 2^2 * 127^2 x^3 + x^2 - 46 x + 103 p 139 p.root 2 exps 8^k d = 139^2 x^3 + x^2 - 50 x - 123 p 151 p.root 6 exps 3^k d = 3^2 * 151^2 x^3 + x^2 - 52 x + 64 p 157 p.root 5 exps 2^k d = 2^4 * 157^2 x^3 + x^2 - 54 x - 169 p 163 p.root 2 exps 5^k d = 163^2 x^3 + x^2 - 60 x - 67 p 181 p.root 2 exps 6^k d = 5^2 * 181^2 x^3 + x^2 - 64 x + 143 p 193 p.root 5 exps 11^k d = 3^2 * 193^2 x^3 + x^2 - 66 x + 59 p 199 p.root 3 exps 12^k d = 5^2 * 199^2 x^3 + x^2 - 70 x - 125 p 211 p.root 2 exps 8^k d = 5^2 * 211^2 x^3 + x^2 - 74 x - 256 p 223 p.root 3 exps 13^k d = 2^2 * 223^2 x^3 + x^2 - 76 x - 212 p 229 p.root 6 exps 2^k d = 2^4 * 229^2 x^3 + x^2 - 80 x + 125 p 241 p.root 7 exps 17^k d = 5^2 * 241^2 x^3 + x^2 - 90 x + 261 p 271 p.root 6 exps 24^k d = 3^2 * 271^2 x^3 + x^2 - 92 x + 236 p 277 p.root 5 exps 2^k d = 2^4 * 277^2 x^3 + x^2 - 94 x + 304 p 283 p.root 3 exps 2^k d = 2^2 * 283^2 x^3 + x^2 - 102 x - 216 p 307 p.root 5 exps 2^k d = 2^2 * 3^2 * 307^2 x^3 + x^2 - 104 x + 371 p 313 p.root 10 exps 7^k d = 313^2 x^3 + x^2 - 110 x - 49 p 331 p.root 3 exps 7^k d = 7^2 * 331^2 x^3 + x^2 - 112 x + 25 p 337 p.root 10 exps 5^k d = 7^2 * 337^2 x^3 + x^2 - 116 x - 517 p 349 p.root 2 exps 6^k d = 349^2 x^3 + x^2 - 122 x + 435 p 367 p.root 6 exps 3^k d = 3^2 * 367^2 x^3 + x^2 - 124 x - 221 p 373 p.root 2 exps 8^k d = 7^2 * 373^2 x^3 + x^2 - 126 x + 365 p 379 p.root 2 exps 8^k d = 5^2 * 379^2 x^3 + x^2 - 132 x - 544 p 397 p.root 5 exps 15^k d = 2^4 * 397^2 x^3 + x^2 - 136 x - 515 p 409 p.root 21 exps 11^k d = 5^2 * 409^2 x^3 + x^2 - 140 x - 343 p 421 p.root 2 exps 8^k d = 7^2 * 421^2 x^3 + x^2 - 144 x - 16 p 433 p.root 5 exps 42^k d = 2^6 * 433^2 x^3 + x^2 - 146 x - 504 p 439 p.root 15 exps 3^k d = 2^2 * 3^2 * 439^2 x^3 + x^2 - 152 x - 220 p 457 p.root 13 exps 5^k d = 2^6 * 457^2 x^3 + x^2 - 154 x + 343 p 463 p.root 3 exps 7^k d = 7^2 * 463^2 x^3 + x^2 - 162 x - 505 p 487 p.root 3 exps 7^k d = 7^2 * 487^2 x^3 + x^2 - 166 x + 536 p 499 p.root 7 exps 2^k d = 2^2 * 3^2 * 499^2 x^3 + x^2 - 174 x - 891 p 523 p.root 2 exps 8^k d = 3^2 * 523^2 x^3 + x^2 - 180 x + 521 p 541 p.root 2 exps 8^k d = 7^2 * 541^2 x^3 + x^2 - 182 x - 81 p 547 p.root 2 exps 8^k d = 3^4 * 547^2 x^3 + x^2 - 190 x - 719 p 571 p.root 3 exps 7^k d = 7^2 * 571^2 x^3 + x^2 - 192 x + 171 p 577 p.root 5 exps 14^k d = 3^4 * 577^2 x^3 + x^2 - 200 x + 512 p 601 p.root 7 exps 31^k d = 2^6 * 601^2 x^3 + x^2 - 202 x - 1169 p 607 p.root 3 exps 6^k d = 607^2 x^3 + x^2 - 204 x + 999 p 613 p.root 2 exps 8^k d = 3^2 * 613^2 x^3 + x^2 - 206 x + 321 p 619 p.root 2 exps 3^k d = 3^4 * 619^2 x^3 + x^2 - 210 x - 1075 p 631 p.root 3 exps 27^k d = 5^2 * 631^2 x^3 + x^2 - 214 x - 1024 p 643 p.root 11 exps 2^k d = 2^2 * 3^2 * 643^2 x^3 + x^2 - 220 x - 1273 p 661 p.root 2 exps 8^k d = 3^2 * 661^2 x^3 + x^2 - 224 x - 997 p 673 p.root 5 exps 10^k d = 7^2 * 673^2 x^3 + x^2 - 230 x + 128 p 691 p.root 3 exps 2^k d = 2^2 * 5^2 * 691^2 x^3 + x^2 - 236 x + 1313 p 709 p.root 2 exps 8^k d = 709^2 x^3 + x^2 - 242 x + 1104 p 727 p.root 5 exps 3^k d = 2^2 * 3^2 * 727^2 x^3 + x^2 - 244 x + 1276 p 733 p.root 6 exps 2^k d = 2^4 * 733^2 x^3 + x^2 - 246 x - 520 p 739 p.root 3 exps 2^k d = 2^2 * 5^2 * 739^2 x^3 + x^2 - 250 x + 1057 p 751 p.root 3 exps 6^k d = 7^2 * 751^2 x^3 + x^2 - 252 x + 729 p 757 p.root 2 exps 8^k d = 3^4 * 757^2 x^3 + x^2 - 256 x - 1481 p 769 p.root 11 exps 7^k d = 5^2 * 769^2 x^3 + x^2 - 262 x - 991 p 787 p.root 2 exps 3^k d = 3^4 * 787^2 x^3 + x^2 - 270 x + 1592 p 811 p.root 3 exps 2^k d = 2^2 * 811^2 x^3 + x^2 - 274 x + 61 p 823 p.root 3 exps 5^k d = 11^2 * 823^2 x^3 + x^2 - 276 x - 307 p 829 p.root 2 exps 7^k d = 11^2 * 829^2 x^3 + x^2 - 284 x + 1011 p 853 p.root 2 exps 5^k d = 3^4 * 853^2 x^3 + x^2 - 286 x - 509 p 859 p.root 2 exps 8^k d = 11^2 * 859^2 x^3 + x^2 - 292 x + 1819 p 877 p.root 2 exps 6^k d = 877^2 x^3 + x^2 - 294 x + 1439 p 883 p.root 2 exps 8^k d = 7^2 * 883^2 x^3 + x^2 - 302 x - 739 p 907 p.root 2 exps 8^k d = 11^2 * 907^2 x^3 + x^2 - 306 x - 1872 p 919 p.root 7 exps 3^k d = 2^2 * 3^2 * 919^2 x^3 + x^2 - 312 x - 2221 p 937 p.root 5 exps 15^k d = 937^2 x^3 + x^2 - 322 x + 1361 p 967 p.root 5 exps 3^k d = 3^4 * 967^2 x^3 + x^2 - 330 x - 2349 p 991 p.root 6 exps 3^k d = 3^2 * 991^2 x^3 + x^2 - 332 x - 480 p 997 p.root 7 exps 2^k d = 2^4 * 3^2 * 997^2 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=  This situation can be summarized as follows (I take the case p = 19 as an example): Let$\zeta$be a$19$-th root of unity then we can form the field of cyclotomic numbers$\mathbb{Q}(\zeta)$. The galois group of this extension (of$\mathbb{Q}$) is the cyclic group$C_{18}$of order$18$. In order to construct a sub-extension$K$of dimension$3$, lets chose a subgroup$H$of$G$of index$3$, i.e. the cyclic subgroup of order$18/3 = 6$. The orbit of$\zeta^2$gives$\{\zeta^2, \zeta^3, \zeta^5, \ldots \text{(complex conjugates)}\}$. The sum$x$of these elements is invariant under$H$so belongs to$K$, so it’s minimal polynomial is cubic :$x^3 + x^2 -6x-7$.The same situation reproduces for the other values of$p$, so the problem rather concerns the powers of a generator in orbits of cosets of subgroups in the automorphism group of the cyclic group$C_p\$.