Problems & Puzzles: Puzzles

Puzzle 1104 Sum of powers of primitive roots Hope you like the following puzzle from Giorgos Kalogeropoulos The primitive roots of prime number 43 are 3, 5, 12, 18, 19, 20, 26, 28, 29, 30, 33 and 34.   3 + 5 + 12 + 18 + 19 + 20 + 26 + 28 + 29 + 30 + 33 + 34 = 257 (prime)   3^2 + 5^2 + 12^2 + 18^2 + 19^2 + 20^2 + 26^2 + 28^2 + 29^2 + 30^2 + 33^2 + 34^2 = 6709 (prime)         3^3 + 5^3 + 12^3 + 18^3 + 19^3 + 20^3 + 26^3 + 28^3 + 29^3 + 30^3 + 33^3 + 34^3 = 188729 (prime)         3^4 + 5^4 + 12^4 + 18^4 + 19^4 + 20^4 + 26^4 + 28^4 + 29^4 + 30^4 + 33^4 + 34^4 = 5527909 (prime)          3^5 + 5^5 + 12^5 + 18^5 + 19^5 + 20^5 + 26^5 + 28^5 + 29^5 + 30^5 + 33^5 + 34^5 = 166291577 (prime)          3^6 + 5^6 + 12^6 + 18^6 + 19^6 + 20^6 + 26^6 + 28^6 + 29^6 + 30^6 + 33^6 + 34^6 = 5098962229 (composite)           If we raise all the primitive roots of 43 to the first 5 powers, the sums of these powers are prime numbers. For any prime p we always use all the primitive roots < p.   Q1. Can you find another prime with the same property? Q2. Can you find a prime whose sums of the first six powers (or more) of its primitive roots are primes?

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During the week 24-30 Sep. 2022, contributions came from Gennady Gusev,

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Gennady wrote:

I have not found a better solution than in the terms of the puzzle.
I tried to find inconsistencies sums. I was looking for the smallest primes p for which the sum of nth powers of primitive roots is prime.
Here are my results:

n       p       sum nth powers of primitive roots of p
1       3       2
2       5       13
3       23      37951
4       5       97
5       11      57383
6       67      219368571593
7       103     546640445276063
8       463     29627950610578593419611
9       719     2355802741378231565358748097
10      1499    3329573544435922302326105706641017
11      1907    97193545286548145942704768713323875181
12      2131    375083361095137837745818538893015779157621
13      1087    70484643716833916143056900417206931233053
14      43      5600271723320002404469
15      647     33817809554056475790959883561701583316455163
16      31      16608062419999808734213
17      3343    49462481776603269621739982857682063586774339058566623220887699
18      4603    64137026318357389663350551520839936412776805869430315659199073098319
19      11959   64778232382706271176297886716710755606603670791416752753478856314236794371861477
20      3343    1563900721213942491118047447510652700054150135217108364249346018699587337
21      6863    62094368348127121913571090691680978621201535598629198065291922362912077724679155007
22      179     126194442164707494403640746066566008589998120827813
23      2347    8569753098717971122028324935157521526518320291258001849475004321943539220022197
24      4603    463053046379188890385405898311170660127699297158054937842086054825752544429380657405053703

Maybe someone wants to continue the table?

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