Math 131
Spring 05
More Number Theory Problems: Conjectures and Patterns
The first four conjectures come from fictional students. There is at least one conjecture that is false, and also at least one that is true. Evaluate each conjecture. Remember, that to prove a conjecture, you need to justify why it will always work. To disprove a conjecture, you only need one counterexample.
Please work on all these problems, and choose one to write up for Monday March 21.
1. Rokia’s Triple Conjecture:
I noticed that, except for 3, 5, and 7, three odd numbers in a row can’t all be prime. For example, if you look at the odds starting with 5: 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, etc. you can see that there are never three primes in a row on this list. I’m not sure why this is true, but I checked it all the way out to 100, and it worked. My conjecture is that for natural numbers greater than 5, there will never be three consecutive odd numbers that are all prime.
2.
I can make bigger primes from smaller primes by writing them next to each other to make one number. It won’t work with 2 or 5, but it works with other primes. For example, 17 and 47 are both prime and 1747 and 4717 are also both prime; 11 and 23 are both prime, and so are 1123 and 2311; 3 and 7 are both prime, and so are 37 and 73. My conjecture is that when you write two primes (besides 2 and 5) next to each other this way, the result is always prime.
3. Palacia’s Prime Formula:
I found a formula that always gives primes. The formula is n2+n+17, where n is any natural number. Here is a table of the first few values; see how the right column is always prime:
n n2+n+17
0 17
1 19
2 23
3 29
4 37
I even tried a few more values, and it keeps working, so it must work. This formula won’t give all the primes that there are, but the right column is always prime.
4. Minai’s Six Pattern:
I made a numbers chart on Excel with only six columns. I noticed that, except for in the first row, all the primes are in the two shaded columns. I don’t know why this is or if it will continue, but I checked for the whole table and it works. My conjecture is that there will never be any primes in columns 2, 3, 4, and 6 (except for the 2 and the 3).
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The next two conjectures are named after real mathematicians. One of them is false, and the other one has never been proven either true or false. Can you tell which is which?
5. Goldbach’s Conjecture:
Every even natural number greater than 2 can be written as the sum of two (not necessarily distinct) primes. For example, 4=2+2, 6=3+3, 8=3+5, 10 = 3+7 or 5+5, etc.
6. de Polignac’s Conjecture:
Every odd natural number greater than 1 can be written as the sum of a power of two and a prime. For example, 3 = 2+1 (1 is a power of 2, and 2 is a prime), 5 = 2+3, 7 = 4+3 or 2+5, 9 = 4+5.
Copyright 2005, Debra K. Borkovitz. You may copy or edit this material for non-profit, educational use only.
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