How can I solve quadratic modular equations?

Asked Jan 03 '23 at 16:06

Active Jan 03 '23 at 16:10

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I looked at this topic How can I solve quadratic equations using modular arithmetic, but it still didn't answer my question.

$$n^2 + n + 8 \equiv 0 \pmod {18}$$

According to the discussion mentioned earlier, we need to complete the square and then solve it. Also, if the linear term is odd, then we can add a multiple of the modulus to make it even. This only works if the modulus is odd. However, in this case modulus is even. How do I solve this equation?

edited Jan 03 '23 at 16:10

lulu

70,402

asked Jan 03 '23 at 16:06

py_math

With such a small modulus, you can simply search. As the modulus is not prime, there could be (and are) more than $2$ solutions. – lulu Jan 03 '23 at 16:08
1

More abstractly, you can factor $18=2\times 3^2$, solve the problem $\pmod 2$ and $\pmod 9$ and then combine those via the Chinese Remainder Theorem. – lulu Jan 03 '23 at 16:08
2

Hint: $4n^2+4n+1=(2n+1)^2$ – cansomeonehelpmeout Jan 03 '23 at 16:11

How can I solve quadratic modular equations?

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