1. Compute . Simplify your answer to a number in the range . Show your work!
2. Prove that for all , is not divisible by 5.
Since every is congruent mod 5 to one of 0, 1, 2, 3, or 4, the table shows that for all n, . Therefore, for all , is not divisible by 5.
3. For what prime numbers p is a perfect square?
Suppose p is prime and , where . Now if and only if , so I may assume x is nonnegative. (If x is negative, then is positive, so I could just work with in what follows.)
If , then I have , or . This is impossible, since p is prime. If , then I have , or . This is impossible, since p is prime.
Thus, I may assume that , so and are both positive.
Since 17 and p are primes and , there are 4 cases.
Case 1. and .
The first equation gives . Plugging this into the second equation, I get . This is okay, since 19 is prime.
Case 2. and .
The second equation gives . Plugging this into the second equation, I get . Since 15 isn't prime, this doesn't work.
Case 3. and .
The second equation gives . This possibility was ruled out earlier.
Case 4. and .
The first equation gives . Plugging this into the second equation, I get . This is a contradiction, since p is a prime number, and the only possible solution is .
The only prime for which is a perfect square is .
4. The Goldbach Conjecture asserts that every even number greater than 2 can be written as the sum of two primes.
(a) Show by specific example that Goldbach's conjecture holds for 158.
(b) Find an even number which can be written as a sum of two primes in at least two different ways.
(c) The Chinese mathematician Chen Jing-Run proved a theorem related to Goldbach's conjecture that is probably the best result known so far. Find out what his result says and state it in your own words.
(c) Chen Jing-Run showed that every sufficiently large even number is the sum of a prime and the product of at most two primes.
5. Find 5 distinct pairs of twin primes.
For example, 3 and 5, 5 and 7, 11 and 13, 29 and 31, and 41 and 43 are twin prime pairs.
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