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Math Help - Number Theory

  1. #1
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    Number Theory

    I want to determine the smallest prime  p such that
    <br /> <br />
\left(\frac {2}{p}\right)=-1<br />

    and the order of [2]_p  is less than  p-1.

    Here <br /> <br />
\left(\frac {2}{p}\right)=-1<br />
    is Legendre symbol.

    I have tried with different prime number but have not got any answer.
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  2. #2
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    p \equiv 3 \ \text{or} \ 5 \ (\text{mod } 8) \ \ \Rightarrow \ \ (2/p) = - 1

    With the help of Wiki's Table of Primitive Roots , the first prime that does not have 2 as its primitive root and is congruent to 3 or 5 is p=13
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  3. #3
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    but the order of [2]_13 is 12, and I am looking for a p where order of 2 is less than p-1?
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  4. #4
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    Quote Originally Posted by peteryellow View Post
    but the order of [2]_13 is 12, and I am looking for a p where order of 2 is less than p-1?
    If you look at the table o_O provided thou shall see that 13 has primitive root 6 and that [2]_{13} = [6]_{13}^5.
    Thus, [2]_{13} is not a primitive root.
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