Proof of inequality - $|ap^2 + bpq + cq^2| \geq 1$ if a,b,c,p,q $\in Z$

Question

I came across an assumption made in the course of a derivation. The assumption states:

If $a,b,c,p,q \in Z$, then $|ap^2 + bpq + cq^2| \geq 1$

However, I feel that we can find values of $a,b,c,p,q \in Z \ni ap^2 + bpq + cq^2 = 0$, which violates the above assumption.

Please note that I attempted a few hit and trials and have not been able to find such set of values for $a,b,c,p,q$.

Can you please help validate or refute my understanding.

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NOTE: Post @Julienne's help, it's important for me to add that the above needs to be seen in the context of the problem statement and not in isolation. The problem statement is as follows:

If $a,b,c \in Z$ and $ax2+bx+c=0$ has an irrational root, then prove than $|f(\lambda)| \geq \frac{1}{q^2}$ , where $\lambda \in (\lambda = \frac pq;p,q \in Z)$ and $f(x)=ax^2+bx+c$.

Answer 1

It suffices to take $p=0$. This reduces the original problem to $|cq^2| \geq 1$. By assumption, $c$ and $q$ are both integers, and $q$ is nonzero. Hence, $|cq^2|$ is atleast 1 provided that $c$ is not zero. Now, $c$ is in fact never zero because the discriminant is not a perfect square.

Answer 2

it appears you have an acceptable answer already. I would state it this way: given $\Delta = b^2 - 4 a c$ is not a perfect square, and $p,q$ are not both zero, then $a p^2 + b pq + c q^2 \neq 0.$

Then the absolute value and everything being integers gives you at least $1$ as the value.

The discussion of this varies a good deal between positive definite (binary) quadratic forms and indefinite forms. For positive forms, there are explicit inequalities. For indefinite forms, the "minimum" is the value of smallest absolute value assumed by the form. The method for finding the minimum of a given form is equivalent to continued fractions, but is due to Lagrange and Gauss. For the calculation I did below for $157 p^2 + 379 pq + 101 q^2,$ we see that there is a $(p,q)$ pair where the value is $-1,$ see lines 32 and 33 of the output below.

The least expensive book that gives a complete discussion of Lagrange-Gauss is Introduction to the Theory of Numbers by Leonard Eugene Dickson, original publication 1929, later issued as paperback reprint, beginning about 1957.

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle   157 379 101

  0  form            157         379         101  delta      3
  1  form            101         227         -71


           0          -1
           1           3

To Return  
           3           1
          -1           0

0  form   101 227 -71   delta  -3
1  form   -71 199 143   delta  1
2  form   143 87 -127   delta  -1
3  form   -127 167 103   delta  2
4  form   103 245 -49   delta  -5
5  form   -49 245 103   delta  2     ambiguous  
6  form   103 167 -127   delta  -1
7  form   -127 87 143   delta  1
8  form   143 199 -71   delta  -3
9  form   -71 227 101   delta  2
10  form   101 177 -121   delta  -1
11  form   -121 65 157   delta  1
12  form   157 249 -29   delta  -9
13  form   -29 273 49   delta  5
14  form   49 217 -169   delta  -1
15  form   -169 121 97   delta  2
16  form   97 267 -23   delta  -11
17  form   -23 239 251   delta  1
18  form   251 263 -11   delta  -24
19  form   -11 265 227   delta  1
20  form   227 189 -49   delta  -4
21  form   -49 203 199   delta  1
22  form   199 195 -53   delta  -4
23  form   -53 229 131   delta  1
24  form   131 33 -151   delta  -1
25  form   -151 269 13   delta  21
26  form   13 277 -67   delta  -4
27  form   -67 259 49   delta  5
28  form   49 231 -137   delta  -1
29  form   -137 43 143   delta  1
30  form   143 243 -37   delta  -7
31  form   -37 275 31   delta  9
32  form   31 283 -1   delta  -283
33  form   -1 283 31   delta  9     ambiguous  
34  form   31 275 -37   delta  -7
35  form   -37 243 143   delta  1
36  form   143 43 -137   delta  -1
37  form   -137 231 49   delta  5
38  form   49 259 -67   delta  -4
39  form   -67 277 13   delta  21
40  form   13 269 -151   delta  -1
41  form   -151 33 131   delta  1
42  form   131 229 -53   delta  -4
43  form   -53 195 199   delta  1
44  form   199 203 -49   delta  -4
45  form   -49 189 227   delta  1
46  form   227 265 -11   delta  -24
47  form   -11 263 251   delta  1
48  form   251 239 -23   delta  -11
49  form   -23 267 97   delta  2
50  form   97 121 -169   delta  -1
51  form   -169 217 49   delta  5
52  form   49 273 -29   delta  -9
53  form   -29 249 157   delta  1
54  form   157 65 -121   delta  -1
55  form   -121 177 101   delta  2
56  form   101 227 -71


  form   101 x^2  + 227 x y  -71 y^2 

minimum was   1rep   x = 675465281141887   y = 2427008577045645 disc 80213 dSqrt 283  M_Ratio  7.851093
Automorph, written on right of Gram matrix:  
88448374521847458527027240500  223406143039383756197594755227
317803104887010695436015074337  802718719168891298764407655099
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$