What is the meaning of this sentence? "The right side of $a\equiv b \pmod c$ is less than or equal to the left side." Let us consider a congruence of the form
$$a\equiv b \pmod c$$
What is the meaning of the following sentence?
$$\text{The right side of the congruence is less than or equal to the left side.}$$

For context, the Jones, et.al, American Mathematical Monthly article &quot;Diophantine Representation of the Set of Prime Numbers&quot; (PDF link via umd.edu) has the following (Lemma 2.4, pg 452; emphasis added):

For any numbers $p$, $n$ and $a\geq 1$ we have the congruence
$$\chi_a(n) \equiv p^n+\psi_a(n)(a-p) \pmod{2ap-p^2-1}$$
Furthermore, when $0&lt; p^n&lt;a$, the right side of the congruence is less than or equal to the left side.
I am not an expert on the topic, but I think you can use the definition of congruence modulo which is that $\displaystyle a\equiv b\ (\bmod c) \ \Leftrightarrow a-b=ck$, so I would think you can replace the equality sign with a greater than or equal to. It would then be $\displaystyle a-b\geqslant ck$. I am not sure, however.
I am not an expert on the topic, but I think you can use the definition of congruence modulo which is that $\displaystyle a\equiv b\ (\bmod c) \ \Leftrightarrow a-b=ck$, so I would think you can replace the equality sign with a greater than or equal to. It would then be $\displaystyle a-b\geqslant ck$. I am not sure, however.