In practice, there are two cases of Legendre symbols which can be computed easily, but are not covered by the previous section. In both cases, they involve LS(

a,p) withafixed, and the primepvarying. Let's look back at an earlier example of Legendre symbols of this type.The value of LS(5,

p) depends only on the congruence class ofpmodulo a specific integer. This is somewhat surprising! In fact, it may not be clear yet that this is the case. Let's compute some Legendre symbols to check it out.The applet below will display a table, with each line providing the remainder of

amodulontogether with the value of LS(a,p), where_{i}pis the_{i}ith prime. The value ofiruns from 2 (thus we start withp_{2}= 3) and ends atB+ 1. In the example below, we seta= 5 andn= 3.Examining the output, we don't see any obvious patterns. Let's try setting

n= 5:This time a pattern does emerge. Upon close inspection, we see that LS(5,

p) = 1 whenpis congruent to 1 or 4 modulo 5, and LS(5,p) = –1 whenpis congruent to 2 or 3 modulo 5.In the next two subsections, we examine two cases similar to the above illustration. For each, your task is to find a pattern similar to the one we spotted for

a= 5.## 11.6.1 Computing LS(–1,

p)Let

pbe an odd prime. We want to know when –1 is congruent to a square modp. As above, use the applet with different values ofnto look for patterns. Here's what we get ifn= 2.That may have not been too helpful! Try other values of

nuntil you find a pattern.

## Research Question 5

For which odd primes

pis LS(–1,p) = 1?

Hint:Once you have a good conjecture, consider Euler's Criterion when searching for a proof.## 11.6.2 Computing LS(2,

p)Same song, second verse. Use the same approach as above to tackle Research Question 6. By now you should be an expert at using the applet, so we leave all of the entries for you to fill in.

## Research Question 6

For which odd primes

pis LS(2,p) = 1?Note:We are primarily looking for a good conjecture for this problem. A proof is fairly difficult.

Section 11.1 | Section 11.2 | Section 11.3 | Section 11.4 | Section 11.5 | Section 11.6

Copyright © 2001 by W. H. Freeman and Company