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