# Are these statements true or false? (Which congruence class does the Jacobi symbol $(\frac {a}{b})$ depend on?)

Statement 1: For non-zero natural numbers $a$, $b$ which are relatively prime, the Jacobi symbol $(\frac {a}{b})$

depends only on the congruence class of $a$ modulo $b$.

I believe this is false because I think it depends on whether or not $a$ and $b$ are $1$ or $3$ mod $4$. Is this reasoning correct?

Statement 2: For non-zero natural numbers $a$, $b$ which are relatively prime, the Jacobi symbol ($\frac {a}{b})$

depends only on the congruence class of $b$ modulo $4a$.

I think this statement is true by the quadratic reciprocity theorem. Is this reasoning correct ?

Neither statement is correct.

Counterexample for the first

$(41,26)$ and $(93,26)$

and for the second

$(7,20)$ and $(7,76)$

• Could you please expand on why they're incorrect then? – user274933 Oct 25 '15 at 22:43
• To be honest, I did not find these counter-examples by hand, but with the PARI/GP-calculator. Just calculate the jacobi-symbols and verify whether they are equal. – Peter Oct 25 '15 at 22:50
• Ok but a Jacobi symbol (a/b) must be dependent on the congruence class of b modulo a, no? – user274933 Oct 25 '15 at 22:55
• Yes, but not only as the counter-examples show. – Peter Oct 25 '15 at 22:56
• I don't see how the counterexamples don't show that to be honest. Could you show me exactly how the counterexample shows otherwise? – user274933 Oct 25 '15 at 22:59