# Young Gauss causing problems with primitive root sums.

Ed Pegg Jr edp at wolfram.com
Fri Jul 14 21:24:40 CEST 2006

```In Article 81 of his /Disquisitiones Arithmeticae/ (1801), Gauss writes
(as translated by A.A. Clarke): "The sum of all primitive roots is
either ≡ 0 (when /p/-1 is divisible by a square), or ≡ ±1 (mod /p/)
(when /p/-1 is the product of unequal prime numbers; if the number of
these is even the sign is positive but if the number is odd, the sign is
negative)."

This seems to be true for all odd primes /p/.

Mod[Sum[PrimitiveRoots[n]], n] - MoebiusMu[n] == 0 is also true for

4, 6, 10, 25, 26, 50, 82, 122, 125, 169, 226, 289, 298, 326, 361, 362,
514, 586, 625, 626, 802, 841, 842, 982

What would these numbers be called? Moebius-Gauss primitive root sum
composites?

Mod[Sum[PrimitiveRoots[n]], n] - MoebiusMu[n] == n/2 for these numbers:

22, 62, 94, 158, 214, 278, 454, 718, 734, 862, 878, 886, 934, 998

It's 1 (or -1) for these numbers:
{38, 74, 106, 194, 274, 314, 338, 346, 386, 398, 458, 466, 542, 562,
578, 614, 674, 706, 758, 778, 898} is 1.
{18, 34, 58, 146, 178, 202, 218, 250, 254, 302, 394, 482, 502, 538, 554,
634, 698, 722, 746, 794, 818, 866, 914, 922, 974} is -1

The sum of primitive roots is irregular for these numbers:

{{2, -2}, {9, -2}, {14, 9}, {27, 20}, {46, 24}, {49, 6},
{54, 49}, {81, 63}, {86, 41}, {98, 7}, {118, 60}, {121, 10},
{134, 65}, {142, 69}, {162, 143}, {166, 85}, {206, 101},
{242, 11}, {243, 189}, {262, 130}, {334, 168}, {343, 42},
{358, 181}, {382, 189}, {422, 213}, {446, 221}, {478, 238},
{486, 431}, {526, 264}, {529, 22}, {566, 281}, {622, 310},
{662, 333}, {686, 41}, {694, 348}, {729, 567}, {766, 384},
{838, 418}, {926, 464}, {958, 481}, {961, 931}}

Unadulterated Primitive Root sums (with 0 for none) are

0, 1, 2, 3, 5, 5, 8, 0, 7, 10, 23, 0, 26, 8, 0, 0, 68, 16, 57, 0, 0,
56, 139, 0, 100, 52, 75, 0, 174, 0, 123, 0, 0, 136, 0, 0, 222,
114, 0, 0, 328, 0, 257, 0, 0, 208, 612, 0, 300, 200, 0, 0, 636,
156, 0, 0, 0, 348, 886, 0, 488, 216, 0, 0, 0, 0, 669, 0, 0, 0,
1064, 0, 876, 444, 0, 0, 0, 0, 1105, 0, 711, 656, 1744, 0, 0,
558, 0, 0, 1780, 0, 0, 0, 0, 988, 0, 0, 1552, 594, 0, 0, 2020,
0, 1853, 0, 0, 1272, 2890, 0, 1962, 0, 0, 0, 2712, 0, 0, 0, 0,
1948, 0, 0, 2309, 976, 0, 0, 2500, 0, 2413, 0, 0, 0, 3536, 0,
0, 1406, 0, 0, 4384, 0, 3335, 0, 0, 1632, 0, 0, 0, 1752, 0, 0,
5364, 0, 3322, 0, 0, 0, 0, 0, 3768, 1974, 0, 0, 0, 1440, 4564,
0, 0, 3736, 7683, 0, 4056, 0, 0, 0, 7266, 0, 0, 0, 0, 3560,
8235, 0, 4344, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8021, 0, 6176, 3104,
0, 0, 8274, 0, 6965, 0, 0, 4040, 0, 0, 0, 3398, 0, 0, 0, 0,
5698, 0, 0, 6100, 0, 0, 0, 3924, 0, 0, 0, 0, 7581, 0, 0, 5424,
13167, 0, 8244, 0, 0, 0, 13048, 0, 0, 0, 0, 0, 12666, 0, 7712,
4608, 6507, 0, 0, 0, 0, 0, 0, 5000, 13052, 0, 0, 4572, 0, 0,
16448, 0, 0, 0, 0, 7466, 18674, 0, 0, 0, 0, 0, 17754, 0, 10569,
0, 0, 8768, 0, 0, 12188, 7088, 0, 0, 13488, 0, 13583, 0, 0, 0,
0, 0, 18496, 0, 0, 0, 21096, 0, 0, 0, 0, 10728, 0, 0

Ed Pegg Jr

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