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\centerline{\bf THEORY OF NUMBERS, Math 115 A}
\centerline{\bf Homework 2 Due Wednesday October 9}
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\begin{enumerate}

\item Use the sieve of Eratosthenes to find all primes less than 250 

\item Find all primes that are difference of the fourth powers of
two integers.

\item Show that there are no ``prime triplets'', that is, primes
     $p,p+2,p+4$, other than 3,5,7. 

\item Let $Q_n=p_1p_2...p_n+1$ where $p_1,\dots,p_n$ are the $n$
smallest primes. Determine the smallest prime factor of $Q_n$ for $n
\leq 6$.  

\item Prove that no polynomial function with integer coefficients can 
take only prime values when evaluated at the integers. HINT: Use the
binomial theorem (i.e the expansion of $(x+y)^p$ in terms of binomial
coefficients) and the fact that a polynomial of degree $d$ cannot
repeat too many times the same value.

\item A prime power is an integer of the form $p^k$, where $p$ is prime
and $n$ is a positive integer greater than 1. Find all prime powers that
differ by 1 and prove that your answer is correct.


\item Find the greatest common divisor of each of the following
sets of integers: (a) 100, 121, (b) 1001, 289 (c) 6,15,21. For the
two items please write the gcd as a linear combination of the
numbers.

\item Show that the gcd of $8a+3$ and $5a+2$ is always one.

\item Every positive integer greater than 6 is the sum of two
relatively prime integers greater than 1.


\end{enumerate}

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