1
$\begingroup$

EDIT: this seems to be a lost cause. Here are the numbers up to 5000 for which I can't get it to work:

ALSO: it turned out to be elementary to show that, for an integer $n$ that is integrally represented as $n = f(w,x,y,z),$ we can always arrange that all $w,x,y,z \geq 0.$

4951  =   4951 ::::: 1  17  22  12      Prime !  
4931  =   4931 ::::: 1  14  20  9      Prime !  
4591  =   4591 ::::: 1  18  28  15      Prime !  
4555  =   5 911 ::::: 1  13  15  6   
4451  =   4451 ::::: 1  23  34  19      Prime !  
4241  =   4241 ::::: 1  14  22  11      Prime !  
4205  =   5 29^2 ::::: 1  13  14  7   
3971  =   11 19^2 ::::: 1  17  23  12   
3881  =   3881 ::::: 1  9  17  12      Prime !  
3875  =   5^3 31 ::::: 1  17  23  14   
3761  =   3761 ::::: 1  22  32  18      Prime !  
3671  =   3671 ::::: 1  35  54  32      Prime !  
3491  =   3491 ::::: 1  31  47  28      Prime !  
3271  =   3271 ::::: 1  28  42  25      Prime !  
3221  =   3221 ::::: 1  14  17  9      Prime !  
3041  =   3041 ::::: 1  11  12  8      Prime !  
2711  =   2711 ::::: 1  22  32  19      Prime !  
2411  =   2411 ::::: 1  7  11  3      Prime !  
2351  =   2351 ::::: 1  9  7  5      Prime !  
2111  =   2111 ::::: 1  23  35  20      Prime !  
1931  =   1931 ::::: 1  23  34  20      Prime !  
1805  =   5 19^2 ::::: 1  11  12  6   
1375  =   5^3 11 ::::: 1  22  33  19   
1181  =   1181 ::::: 1  12  15  8      Prime !  
125  =   5^3 ::::: 1  7  8  4 

On page 2 of PDF the norm form for $e^{2 \pi i/5}$ is given, namely $$ f(w,x,y,z) = \det( w I + x A + y A^2 + z A^3), $$ where $$ A = \left( \begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & -1 & -1 & -1 \end{array} \right) $$ or

x^4 + (-w + (-y - z))*x^3 + (w^2 + (-3*y + 2*z)*w + (y^2 + 2*z*y + z^2))*x^2 + (-w^3 + (2*y + 2*z)*w^2 + (2*y^2 - z*y - 3*z^2)*w + (-y^3 - 3*z*y^2 + 2*z^2*y - z^3))*x + (w^4 + (-y - z)*w^3 + (y^2 - 3*z*y + z^2)*w^2 + (-y^3 + 2*z*y^2 + 2*z^2*y - z^3)w + (y^4 - zy^3 + z^2*y^2 - z^3*y + z^4))

Using the minimal polynomial of $A,$ we can show that, given two integer quadruples $(s,t,u,v)$ and $(w,x,y,z),$ there is another integer quadruple $(o,p,q,r)$ such that $$ ( s I + t A + u A^2 + v A^3) ( w I + x A + y A^2 + z A^3) =( o I + p A + q A^2 + r A^3), $$ so that $$ f(s,t,u,v) f(w,x,y,z) = f(o,p,q,r). $$ I am less clear on the following things:

Question: it seems every number represented by at least one integer quadruple $(w,x,y,z)$ can be represented with a quadruple such that $w=0.$ This is both the substance of the question Show that each number $n$ whose divisors $d$ $=$ $0, 1$ $\pmod 5$ has the form $P(x)$ and the sense of my computer experiments.

n    factor      w  x  y  z
1  =    1  ::::: 0  0  0  1
1  =    1  ::::: 0  0  1  0
1  =    1  ::::: 0  0  1  1
1  =    1  ::::: 0  1  0  0
1  =    1  ::::: 0  1  0  1
1  =    1  ::::: 0  1  1  0
1  =    1  ::::: 0  1  1  1
1  =    1  ::::: 0  1  2  1
1  =    1  ::::: 0  13  21  13
1  =    1  ::::: 0  2  3  2
1  =    1  ::::: 0  3  5  3
1  =    1  ::::: 0  5  8  5
1  =    1  ::::: 0  8  13  8
5  =   5 ::::: 0  1  2  2
5  =   5 ::::: 0  2  2  1
11  =   11 ::::: 0  0  1  2
11  =   11 ::::: 0  0  2  1
11  =   11 ::::: 0  1  0  2
11  =   11 ::::: 0  1  1  2
11  =   11 ::::: 0  1  2  0
11  =   11 ::::: 0  1  3  2
11  =   11 ::::: 0  2  0  1
11  =   11 ::::: 0  2  1  0
11  =   11 ::::: 0  2  1  1
11  =   11 ::::: 0  2  3  1
11  =   11 ::::: 0  2  4  3
11  =   11 ::::: 0  3  4  2
25  =   5^2 ::::: 0  11  18  11
25  =   5^2 ::::: 0  1  3  1
25  =   5^2 ::::: 0  2  1  2
25  =   5^2 ::::: 0  3  4  3
25  =   5^2 ::::: 0  4  7  4
25  =   5^2 ::::: 0  7  11  7
31  =   31 ::::: 0  1  3  3
31  =   31 ::::: 0  2  3  3
31  =   31 ::::: 0  3  3  1
31  =   31 ::::: 0  3  3  2
31  =   31 ::::: 0  3  6  4
31  =   31 ::::: 0  4  6  3
41  =   41 ::::: 0  1  2  3
41  =   41 ::::: 0  3  2  1
41  =   41 ::::: 0  3  5  4
41  =   41 ::::: 0  4  5  3
41  =   41 ::::: 0  4  7  5
41  =   41 ::::: 0  5  7  4
55  =   5 11 ::::: 0  0  2  3
55  =   5 11 ::::: 0  0  3  2
55  =   5 11 ::::: 0  1  1  3
55  =   5 11 ::::: 0  2  0  3
55  =   5 11 ::::: 0  2  3  0
55  =   5 11 ::::: 0  2  5  3
55  =   5 11 ::::: 0  3  0  2
55  =   5 11 ::::: 0  3  1  1
55  =   5 11 ::::: 0  3  2  0
55  =   5 11 ::::: 0  3  5  2
55  =   5 11 ::::: 0  5  9  6
55  =   5 11 ::::: 0  6  9  5
61  =   61 ::::: 0  0  1  3
61  =   61 ::::: 0  0  3  1
61  =   61 ::::: 0  1  0  3
61  =   61 ::::: 0  1  3  0
61  =   61 ::::: 0  1  4  3
61  =   61 ::::: 0  2  2  3
61  =   61 ::::: 0  3  0  1
61  =   61 ::::: 0  3  1  0
61  =   61 ::::: 0  3  2  2
61  =   61 ::::: 0  3  4  1
71  =   71 ::::: 0  1  4  2
71  =   71 ::::: 0  2  1  3
71  =   71 ::::: 0  2  4  1
71  =   71 ::::: 0  2  5  4
71  =   71 ::::: 0  3  1  2
71  =   71 ::::: 0  4  5  2
101  =   101 ::::: 0  4  8  5
101  =   101 ::::: 0  5  8  4
101  =   101 ::::: 0  7  12  8
101  =   101 ::::: 0  8  12  7
121  =   11^2 ::::: 0  12  19  12
121  =   11^2 ::::: 0  1  3  4
121  =   11^2 ::::: 0  1  4  1
121  =   11^2 ::::: 0  14  23  14
121  =   11^2 ::::: 0  1  4  4
121  =   11^2 ::::: 0  2  5  2
121  =   11^2 ::::: 0  3  1  3
121  =   11^2 ::::: 0  3  2  3
121  =   11^2 ::::: 0  3  4  4
121  =   11^2 ::::: 0  3  6  5
121  =   11^2 ::::: 0  4  3  1
121  =   11^2 ::::: 0  4  4  1
121  =   11^2 ::::: 0  4  4  3
121  =   11^2 ::::: 0  4  5  4
121  =   11^2 ::::: 0  5  6  3
121  =   11^2 ::::: 0  5  7  5
121  =   11^2 ::::: 0  5  9  5
121  =   11^2 ::::: 0  6  10  7
121  =   11^2 ::::: 0  7  10  6
121  =   11^2 ::::: 0  7  12  7
121  =   11^2 ::::: 0  9  14  9
131  =   131 ::::: 0  2  3  4
131  =   131 ::::: 0  4  3  2
131  =   131 ::::: 0  6  11  7
131  =   131 ::::: 0  7  11  6
151  =   151 ::::: 0  1  2  4
151  =   151 ::::: 0  4  2  1
151  =   151 ::::: 0  5  8  6
151  =   151 ::::: 0  6  8  5
151  =   151 ::::: 0  8  14  9
151  =   151 ::::: 0  9  14  8
155  =   5 31 ::::: 0  3  7  5
155  =   5 31 ::::: 0  4  6  5
155  =   5 31 ::::: 0  5  6  4
155  =   5 31 ::::: 0  5  7  3
181  =   181 ::::: 0  0  3  4
181  =   181 ::::: 0  0  4  3
181  =   181 ::::: 0  1  1  4
181  =   181 ::::: 0  3  0  4
181  =   181 ::::: 0  3  4  0
181  =   181 ::::: 0  3  7  4
181  =   181 ::::: 0  4  0  3
181  =   181 ::::: 0  4  1  1
181  =   181 ::::: 0  4  3  0
181  =   181 ::::: 0  4  7  3
191  =   191 ::::: 0  10  17  11
191  =   191 ::::: 0  11  17  10
191  =   191 ::::: 0  1  5  3
191  =   191 ::::: 0  2  1  4
191  =   191 ::::: 0  3  5  1
191  =   191 ::::: 0  4  1  2
205  =   5 41 ::::: 0  0  1  4
205  =   5 41 ::::: 0  0  4  1
205  =   5 41 ::::: 0  1  0  4
205  =   5 41 ::::: 0  1  4  0
205  =   5 41 ::::: 0  1  5  4
205  =   5 41 ::::: 0  3  3  4
205  =   5 41 ::::: 0  4  0  1
205  =   5 41 ::::: 0  4  1  0
205  =   5 41 ::::: 0  4  3  3
205  =   5 41 ::::: 0  4  5  1
211  =   211 ::::: 0  10  15  9
211  =   211 ::::: 0  2  5  5
211  =   211 ::::: 0  3  5  5
211  =   211 ::::: 0  5  5  2
211  =   211 ::::: 0  5  5  3
211  =   211 ::::: 0  9  15  10
241  =   241 ::::: 0  2  6  3
241  =   241 ::::: 0  2  6  5
241  =   241 ::::: 0  3  1  4
241  =   241 ::::: 0  3  6  2
241  =   241 ::::: 0  4  1  3
241  =   241 ::::: 0  5  6  2
251  =   251 ::::: 0  1  5  2
251  =   251 ::::: 0  2  5  1
251  =   251 ::::: 0  3  2  4
251  =   251 ::::: 0  4  2  3
271  =   271 ::::: 0  2  4  5
271  =   271 ::::: 0  4  7  6
271  =   271 ::::: 0  5  4  2
271  =   271 ::::: 0  6  7  4
281  =   281 ::::: 0  5  10  6
281  =   281 ::::: 0  5  10  7
281  =   281 ::::: 0  6  10  5
281  =   281 ::::: 0  7  10  5
305  =   5 61 ::::: 0  12  20  13
305  =   5 61 ::::: 0  13  20  12
305  =   5 61 ::::: 0  1  4  5
305  =   5 61 ::::: 0  5  4  1
305  =   5 61 ::::: 0  8  13  9
305  =   5 61 ::::: 0  9  13  8
311  =   311 ::::: 0  5  9  7
311  =   311 ::::: 0  7  9  5
331  =   331 ::::: 0  3  7  6
331  =   331 ::::: 0  4  9  6
331  =   331 ::::: 0  6  7  3
331  =   331 ::::: 0  6  9  4
341  =   11 31 ::::: 0  11  19  12
341  =   11 31 ::::: 0  12  19  11
341  =   11 31 ::::: 0  13  22  14
341  =   11 31 ::::: 0  1  3  5
341  =   11 31 ::::: 0  14  22  13
341  =   11 31 ::::: 0  1  5  5
341  =   11 31 ::::: 0  3  4  5
341  =   11 31 ::::: 0  4  5  5
341  =   11 31 ::::: 0  5  3  1
341  =   11 31 ::::: 0  5  4  3
341  =   11 31 ::::: 0  5  5  1
341  =   11 31 ::::: 0  5  5  4
341  =   11 31 ::::: 0  7  13  8
341  =   11 31 ::::: 0  8  13  7
355  =   5 71 ::::: 0  10  16  9
355  =   5 71 ::::: 0  2  3  5
355  =   5 71 ::::: 0  5  3  2
355  =   5 71 ::::: 0  6  11  8
355  =   5 71 ::::: 0  8  11  6
355  =   5 71 ::::: 0  9  16  10
361  =   19^2 ::::: 0  10  17  10
361  =   19^2 ::::: 0  11  17  11
361  =   19^2 ::::: 0  1  5  1
361  =   19^2 ::::: 0  3  7  3
361  =   19^2 ::::: 0  4  1  4
361  =   19^2 ::::: 0  4  3  4
361  =   19^2 ::::: 0  5  6  5
361  =   19^2 ::::: 0  6  11  6
361  =   19^2 ::::: 0  7  10  7
401  =   401 ::::: 0  1  2  5
401  =   401 ::::: 0  5  2  1
401  =   401 ::::: 0  7  11  8
401  =   401 ::::: 0  8  11  7
421  =   421 ::::: 0  0  3  5
421  =   421 ::::: 0  0  5  3
421  =   421 ::::: 0  15  25  16
421  =   421 ::::: 0  16  25  15
421  =   421 ::::: 0  2  2  5
421  =   421 ::::: 0  3  0  5
421  =   421 ::::: 0  3  5  0
421  =   421 ::::: 0  3  8  5
421  =   421 ::::: 0  5  0  3
421  =   421 ::::: 0  5  2  2
421  =   421 ::::: 0  5  3  0
421  =   421 ::::: 0  5  8  3
431  =   431 ::::: 0  5  7  6
431  =   431 ::::: 0  6  7  5
451  =   11 41 ::::: 0  0  2  5
451  =   11 41 ::::: 0  0  5  2
451  =   11 41 ::::: 0  11  18  12
451  =   11 41 ::::: 0  12  18  11
451  =   11 41 ::::: 0  1  6  4
451  =   11 41 ::::: 0  2  0  5
451  =   11 41 ::::: 0  2  1  5
451  =   11 41 ::::: 0  2  5  0
451  =   11 41 ::::: 0  2  7  5
451  =   11 41 ::::: 0  3  3  5
451  =   11 41 ::::: 0  3  8  6
451  =   11 41 ::::: 0  4  6  1
451  =   11 41 ::::: 0  5  0  2
451  =   11 41 ::::: 0  5  1  2
451  =   11 41 ::::: 0  5  2  0
451  =   11 41 ::::: 0  5  3  3
451  =   11 41 ::::: 0  5  7  2
451  =   11 41 ::::: 0  6  8  3
451  =   11 41 ::::: 0  6  9  7
451  =   11 41 ::::: 0  7  13  9
451  =   11 41 ::::: 0  7  9  6
451  =   11 41 ::::: 0  9  13  7
461  =   461 ::::: 0  0  4  5
461  =   461 ::::: 0  0  5  4
461  =   461 ::::: 0  1  1  5
461  =   461 ::::: 0  4  0  5
461  =   461 ::::: 0  4  5  0
461  =   461 ::::: 0  4  9  5
461  =   461 ::::: 0  5  0  4
461  =   461 ::::: 0  5  1  1
461  =   461 ::::: 0  5  4  0
461  =   461 ::::: 0  5  9  4
491  =   491 ::::: 0  2  7  4
491  =   491 ::::: 0  3  1  5
491  =   491 ::::: 0  4  7  2
491  =   491 ::::: 0  5  1  3
505  =   5 101 ::::: 0  1  6  3
505  =   5 101 ::::: 0  3  2  5
505  =   5 101 ::::: 0  3  6  1
505  =   5 101 ::::: 0  4  9  7
505  =   5 101 ::::: 0  5  2  3
505  =   5 101 ::::: 0  7  9  4
521  =   521 ::::: 0  0  1  5
521  =   521 ::::: 0  0  5  1
521  =   521 ::::: 0  1  0  5
521  =   521 ::::: 0  1  5  0
521  =   521 ::::: 0  1  6  5
521  =   521 ::::: 0  4  4  5
521  =   521 ::::: 0  5  0  1
521  =   521 ::::: 0  5  1  0
521  =   521 ::::: 0  5  4  4
521  =   521 ::::: 0  5  6  1
541  =   541 ::::: 0  2  5  6
541  =   541 ::::: 0  4  8  7
541  =   541 ::::: 0  6  5  2
541  =   541 ::::: 0  7  8  4
571  =   571 ::::: 0  14  23  15
571  =   571 ::::: 0  15  23  14
571  =   571 ::::: 0  3  5  6
571  =   571 ::::: 0  6  5  3
605  =   5 11^2 ::::: 0  2  7  6
605  =   5 11^2 ::::: 0  3  8  4
605  =   5 11^2 ::::: 0  4  1  5
605  =   5 11^2 ::::: 0  4  8  3
605  =   5 11^2 ::::: 0  5  1  4
605  =   5 11^2 ::::: 0  5  8  7
605  =   5 11^2 ::::: 0  6  7  2
605  =   5 11^2 ::::: 0  7  8  5
641  =   641 ::::: 0  1  6  2
641  =   641 ::::: 0  2  6  1
641  =   641 ::::: 0  4  3  5
641  =   641 ::::: 0  5  3  4
655  =   5 131 ::::: 0  6  12  7
655  =   5 131 ::::: 0  7  12  6
661  =   661 ::::: 0  7  12  9
661  =   661 ::::: 0  9  12  7
671  =   11 61 ::::: 0  10  15  8
671  =   11 61 ::::: 0  10  16  11
671  =   11 61 ::::: 0  11  16  10
671  =   11 61 ::::: 0  1  5  6
671  =   11 61 ::::: 0  2  7  3
671  =   11 61 ::::: 0  3  7  2
671  =   11 61 ::::: 0  4  2  5
671  =   11 61 ::::: 0  5  11  7
671  =   11 61 ::::: 0  5  2  4
671  =   11 61 ::::: 0  6  5  1
671  =   11 61 ::::: 0  7  11  5
671  =   11 61 ::::: 0  8  15  10
691  =   691 ::::: 0  14  24  15
691  =   691 ::::: 0  1  4  6
691  =   691 ::::: 0  15  24  14
691  =   691 ::::: 0  5  10  8
691  =   691 ::::: 0  6  4  1
691  =   691 ::::: 0  8  10  5
701  =   701 ::::: 0  11  16  9
701  =   701 ::::: 0  9  16  11
751  =   751 ::::: 0  3  4  6
751  =   751 ::::: 0  6  4  3
755  =   5 151 ::::: 0  4  5  6
755  =   5 151 ::::: 0  6  5  4
761  =   761 ::::: 0  3  8  7
761  =   761 ::::: 0  7  8  3
775  =   5^2 31 ::::: 0  1  3  6
775  =   5^2 31 ::::: 0  6  3  1
781  =   11 71 ::::: 0  12  21  13
781  =   11 71 ::::: 0  13  21  12
781  =   11 71 ::::: 0  1  6  6
781  =   11 71 ::::: 0  2  3  6
781  =   11 71 ::::: 0  3  7  7
781  =   11 71 ::::: 0  4  7  7
781  =   11 71 ::::: 0  5  11  8
781  =   11 71 ::::: 0  5  6  6
781  =   11 71 ::::: 0  6  3  2
781  =   11 71 ::::: 0  6  6  1
781  =   11 71 ::::: 0  6  6  5
781  =   11 71 ::::: 0  7  7  3
781  =   11 71 ::::: 0  7  7  4
781  =   11 71 ::::: 0  8  11  5
781  =   11 71 ::::: 0  8  15  9
781  =   11 71 ::::: 0  9  15  8
811  =   811 ::::: 0  4  10  7
811  =   811 ::::: 0  7  10  4
821  =   821 ::::: 0  10  18  11
821  =   821 ::::: 0  11  18  10
841  =   29^2 ::::: 0  13  20  13
841  =   29^2 ::::: 0  13  22  13
841  =   29^2 ::::: 0  1  6  1
841  =   29^2 ::::: 0  4  9  4
841  =   29^2 ::::: 0  5  1  5
841  =   29^2 ::::: 0  5  4  5
841  =   29^2 ::::: 0  6  7  6
841  =   29^2 ::::: 0  7  13  7
841  =   29^2 ::::: 0  9  13  9
881  =   881 ::::: 0  10  14  9
881  =   881 ::::: 0  1  2  6
881  =   881 ::::: 0  6  2  1
881  =   881 ::::: 0  9  14  10
905  =   5 181 ::::: 0  7  14  9
905  =   5 181 ::::: 0  9  14  7
911  =   911 ::::: 0  13  21  14
911  =   911 ::::: 0  14  21  13
941  =   941 ::::: 0  1  7  5
941  =   941 ::::: 0  2  1  6
941  =   941 ::::: 0  5  7  1
941  =   941 ::::: 0  6  1  2
955  =   5 191 ::::: 0  2  8  5
955  =   5 191 ::::: 0  3  1  6
955  =   5 191 ::::: 0  5  8  2
955  =   5 191 ::::: 0  6  1  3
961  =   31^2 ::::: 0  11  19  11
961  =   31^2 ::::: 0  16  25  16
961  =   31^2 ::::: 0  1  7  4
961  =   31^2 ::::: 0  2  7  2
961  =   31^2 ::::: 0  3  2  6
961  =   31^2 ::::: 0  3  6  7
961  =   31^2 ::::: 0  3  8  3
961  =   31^2 ::::: 0  4  7  1
961  =   31^2 ::::: 0  5  2  5
961  =   31^2 ::::: 0  5  3  5
961  =   31^2 ::::: 0  5  9  8
961  =   31^2 ::::: 0  6  2  3
961  =   31^2 ::::: 0  7  6  3
961  =   31^2 ::::: 0  7  9  7
961  =   31^2 ::::: 0  8  11  8
961  =   31^2 ::::: 0  8  9  5
961  =   31^2 ::::: 0  9  16  9
971  =   971 ::::: 0  6  8  7
971  =   971 ::::: 0  7  8  6
991  =   991 ::::: 0  0  5  6
991  =   991 ::::: 0  0  6  5
991  =   991 ::::: 0  1  1  6
991  =   991 ::::: 0  5  0  6
991  =   991 ::::: 0  5  11  6
991  =   991 ::::: 0  5  6  0
991  =   991 ::::: 0  6  0  5
991  =   991 ::::: 0  6  1  1
991  =   991 ::::: 0  6  11  5
991  =   991 ::::: 0  6  5  0
n      factor      w  x  y  z
$\endgroup$
  • 1
    $\begingroup$ What does "apparently we can demand" mean? $\endgroup$ – franz lemmermeyer May 15 '17 at 4:40
  • $\begingroup$ @franzlemmermeyer, edited, is this better? $\endgroup$ – Will Jagy May 15 '17 at 15:22
  • $\begingroup$ So it is wishful thinking after having computed five examples? $\endgroup$ – franz lemmermeyer May 15 '17 at 16:29
  • 1
    $\begingroup$ You are looking at the integers $n \in \mathbb{Z}$ such that there exists $\alpha \in \mathcal{O}_K, K= \mathbb{Q}(\zeta_5)$ with $N(\alpha) = n$. And you are computing the first few coefficients of the incomplete Dedekind zeta function $\sum_{I \subset \mathcal{O}_K, I \sim (1)} N(I)^{-s}$ where $I$ ranges over the principal ideals of $\mathcal{O}_K$. $\endgroup$ – reuns May 15 '17 at 23:47
  • $\begingroup$ What do you mean with "completely multiplicative" and "primes $\equiv 1\bmod 5$" ? You are not looking at the complete Dedekind zeta function $\zeta_K(s)=\sum_{I \subset \mathcal{O}_K} N(I)^{-s}$ (all ideals of $\mathcal{O}_K$, not only the principal ones) so you don't have an Euler product and you don't have the cyclotomic zeta result $\zeta_K(s) = \prod_{\chi \bmod 5} L(s,\chi)$ $\endgroup$ – reuns May 15 '17 at 23:49

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