Given a quadratic polynomial ax^2+bx+c, with a, b and c being integers, is there a characterization of all primes p for which the equation ax^2+bx+c≡0(modp) has solutions?
Bayobusalue
Answered question
2022-11-03
Primes modulo which a given quadratic equation has roots Given a quadratic polynomial , with a, b and c being integers, is there a characterization of all primes p for which the equation
has solutions?
Answer & Explanation
Savanna Smith
Beginner2022-11-04Added 17 answers
Step 1 I never noticed this one before.
has one root for odd primes p with .
has three distinct roots for odd p with and in integers.
has no roots for odd p with and in integers (not necessarily positive integers). Here we go, no roots (mod2), but a doubled root and a single (mod23), as
Step 2 Strange but true. Easy to confirm by computer for primes up to 1000, say. The example you can see completely proved in books, Ireland and Rosen for example, is , often with the phrase "the cubic character of 2" and the topic "cubic reciprocity." 2 is a cube for primes and any prime Also, 2 is a cube for primes and in integers. However, 2 is not a cube for primes and in integers. (Gauss) 3 is a cube for primes and any prime Also, 3 is a cube for primes and in integers. However, 3 is not a cube for primes and in integers. (Jacobi)