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Higher Arithmetic: An Algorithmic Introduction to Number Theory

Harold M. Edwards

Chapter 15

Tables of Indices mod $p$ - all with Video Answers

Educators


Chapter Questions

Problem 1

Jacobi's tables for the prime 31 are based on the primitive root $17 \bmod 31$. Construct these tables.

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Problem 2

Choose a prime $p$, find a primitive root $\bmod p$, and construct the corresponding tables.

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04:34

Problem 3

Modulo one of the primes 151 and 157 the polynomial $x^2+$ $2 x-1$ has two roots, and modulo the other it has none. Use Jacobi's tables to determine which is which, and find the roots in the case in which there are roots. Verify that $x^2+2 x-1 \equiv\left(x-r_1\right)\left(x-r_2\right) \bmod p$, where $p$ is the prime for which there are two roots and $r_1$ and $r_2$ are the roots.

Gabriel Eduok
Gabriel Eduok
Numerade Educator

Problem 4

Use Jacobi's table to find three primitive roots mod 157. How many primitive roots are there mod 157 ? How many mod 151?

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00:45

Problem 5

Find a number that has no cube root mod 157. Find a number that does have a cube root mod 157 , and find all of its cube roots. Verify by cubing on your calculator and checking that the cube minus the number is divisible by 157 .

Chandler Austin
Chandler Austin
Numerade Educator
04:28

Problem 6

Find the order of $30 \bmod 157$. Find the order of $90 \bmod 157$.

James Chok
James Chok
Numerade Educator

Problem 7

Use the tables to compute a million (that is, $\left.10^6\right) \bmod 151$ and mod 157. (You can find both answers using mental calculation alone.) Check your answer using a calculator.

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