I have the following problem: I have a diophantine equation of the form: [math]f(x,a,b)+c=0\tag{**}[/math] that has a solution and by reducing it modulo [math]p^n[/math] I get a congruence of the form
[math]h(x,a,b)+c\equiv 0\pmod {p^n}[/math]where [math]x[/math] is the **variable** and [math]a,b,c[/math] are **parameters** and [math]p≥2[/math] is a prime and [math]n[/math] is a positive integer. The above congruence has also solutions.
Let [math]g(a,b,c)=a+b-2c[/math] be a quantity computed **only** from [math]a,b,c[/math]. In my context I want to see if [math]g(a,b,c)[/math] has a **specific form**. In the [math]a,b,c[/math] formula, this form is not clear. However, If I replace [math]c[/math] from [math](**)[/math] I get the desired result. Then my question is about the existence of any obstruction related to this operation: Replacing a variable in an expression of parameters.
[math]h(x,a,b)+c\equiv 0\pmod {p^n}[/math]where [math]x[/math] is the **variable** and [math]a,b,c[/math] are **parameters** and [math]p≥2[/math] is a prime and [math]n[/math] is a positive integer. The above congruence has also solutions.
Let [math]g(a,b,c)=a+b-2c[/math] be a quantity computed **only** from [math]a,b,c[/math]. In my context I want to see if [math]g(a,b,c)[/math] has a **specific form**. In the [math]a,b,c[/math] formula, this form is not clear. However, If I replace [math]c[/math] from [math](**)[/math] I get the desired result. Then my question is about the existence of any obstruction related to this operation: Replacing a variable in an expression of parameters.