Solutions to Problem Set 8

Math 345/504

2-12-2018

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.

Now gives

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 .

[MATH 504]

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.

(a) .

(b) .

(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.

We spend our time envying people whom we wouldn't like to be. - Jean Rostand

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