Sums of Legendre Symbols Question

[doubleaxel195]

Proposition:

$$\sum_{i=0}^{p-1} (\frac{i^2+a}{p})=-1$$ for any odd prime p and any integer a. (I am referring to the Legendre Symbol).

I was reading a paper where they claimed it was true for the a=1 case and referred to a source that I don't have immediate access to. So I was wondering if anyone knows if this is true or a source that talks about this? I know it doesn't mean it's necessarily true, but this proposition has been true with all the examples I've looked at with Mathematica. Thanks!

 

[doubleaxel195]

Nevermind, I found this in the exercises of Ireland and Rosen on page 63. Just thought I would post where I found it in case anyone else needs to know.

 

[RamaWolf]

In your formula, the integer a should not be zero, because

$$\sum_{i=0}^{p-1} (\frac{i^2}{p})=p-1$$

(Note: $$(\frac{0}{p}) = 0$$ for p > 2 and
$$(\frac{x^2}{p})= 1$$ for gcd(x,p)=1)

 

[DonAntonio]

In your formula, the integer a should not be zero, because

$$\sum_{i=0}^{p-1} (\frac{i^2}{p})=p-1$$

(Note: $$(\frac{0}{p}) = 0$$ for p > 2 and
$$(\frac{x^2}{p})= 1$$ for gcd(x,p)=1)

Even if the integer is zero modulo p the formula works, since $\,p-1=-1\pmod p$

DonAntonio

 

[blue_raver22]

I would first commend the individual for seeking clarification and additional sources to support their understanding of the proposition. In mathematics and science, it is always important to verify claims and seek out evidence to support them.

In this case, the proposition appears to be true based on the examples provided and the fact that it is a well-known property of Legendre symbols. However, I would suggest consulting a more comprehensive source, such as a textbook on number theory or a reputable mathematical journal, to confirm the validity of the statement.

Furthermore, it is important to note that while the proposition may hold true for all the examples examined, it does not necessarily mean that it is universally true for all cases. It is always possible that there may be counterexamples or exceptions that have not been considered. Therefore, it is important to continue researching and seeking out evidence to support or refute the proposition.

In conclusion, I would encourage the individual to continue their investigation and consult additional sources to gain a deeper understanding of this proposition. It is through critical thinking and thorough research that we can confidently make claims and contribute to the advancement of scientific knowledge.