Polynomial Congruence/Examples/Arbitrary Example 3

Example of Polynomial Congruence

The polynomial congruence:

$x^2 \equiv 1 \pmod 8$

has solutions:

$x \in \set {1, 3, 5, 7} \pmod 8$


Proof

\(\ds x^2\) \(\equiv\) \(\ds 1\) \(\ds \pmod 8\)
\(\ds x^2 - 1\) \(\equiv\) \(\ds 0\) \(\ds \pmod 8\)
\(\ds \leadsto \ \ \) \(\ds \paren {x + 1} \paren {x - 1}\) \(\equiv\) \(\ds 0\) \(\ds \pmod 8\)
\(\ds \leadsto \ \ \) \(\ds \paren {x - 3} \paren {x - 5}\) \(\equiv\) \(\ds 0\) \(\ds \pmod 8\)


This needs considerable tedious hard slog to complete it.
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Sources

  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): congruence equation
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): congruence equation
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