Prime Numbers: Pattern Hunters
Mathletes are always hunting for patterns in prime numbers. These are the whole numbers that have exactly two factors: 1 and itself. For example: 5 is prime because it only has two factors: 1 and 5; 1 x 5 = 5. 6 is not prime (it is composite) because it has more than just two factors (1 and 6); it also has 2 and 3; 2 x 3 = 6.
Understanding primes was once thought to be a useless exercise for mathematicians but now they are considered critical for securing the internet and countless other applications. I understand primes as they relate to rectangles. Prime numbers can make rectangles of only 1 by the number. In other words, if you take tennis balls and try to make a rectangle of 2 by something or 3 x something or 5 by something, you can only do this with composite numbers like 4, 6, 8, 9, 10, 12, 14, 15, and so on.
When exploring these properties, the children jumped at 9, 15, and 21 as being prime until they saw that they could be made into a rectangle of 3x3, or 3x5, and 3x7, respectively.
When they looked at 2,3,5,7,11,13,17,19, and so on, they could only make rectangles of 1 by that number.
We made a connection from last week's lesson of highlighting all of the multiples by me giving them a page of numbers from 1-500 and 501-1000 with all of the composite numbers highlighted in red. What was left unhighlighted? The prime numbers of course.
The Abel Prize and the Fields Metal have been offered to anyone who could come up with a pattern in prime numbers that would predict the next prime. The prize is worth about $1,000,000. I have been looking for patterns in primes for the last 45 years and will likely continue for another 45 years, or at least 43 since that is a prime number. The children were intrigued.
This week they will be exploring prime number connections and try to find the pattern in primes.
For the younger children, they should concentrate on only understanding the first nine prime numbers (2,3,5,7,11,13,17,19, and 23). Incidentally, many of the children were able to see that the sum of the first nine primes is 100. The other activity that should be attainable by the K-1 kids is looking for the differences of the primes. For example, the difference between 2 and 3 is 1, between 3 and 5 is 2, between 5 and 7 is 2, but the difference between 7 and 11 is 4 and so on.
The attached pdf is an enlarged version of the first 170 prime numbers to make finding differences easier. Some children can look at 17 and 19 and see a difference of two; some will subtract 7 from 9 to get two; and the youngest of our children will actually use their fingers to count up from 17: start with 18, then 19 (how many fingers?)=2. Try a more difficult one like 23 and 29; count up from 23 so we start with 24, 25, 26, 27, 28, and 29 (six fingers). This is an important discipline. Finger counting is very healthy.
For the older kids, they should develop mental math strategies for finding differences like looking at 9-3=6 in the 23 to 29 example above.
Some other great activities on the pdfs are looking at the number of primes in each column and row. For example, there are 22 prime numbers in the column ending in ones from 1-500. There are 4 primes in the row from 1-10. Look for patterns in these results.
Many mathematicians have looked for patterns in the number of primes in the first one hundred numbers (25), then the number of primes from 101-200 (21) and so on. The children can do this up to 1000.
One of the most challenging activities is looking for TWIN PRIMES. These are prime numbers with a difference of 2. For example, 3 and 5, 5 and 7, 11 and 13, 17 and 19. There are dozens of these in the first 170 primes.
There are two other activities that are quite challenging. One is to find consecutive primes (consecutive means one after another) that are 10 apart. The first one is 139 and 149. The children might say 3 and 13, but this is not such a pair because there are prime numbers in between 3 and 13 like 5, 7, and 11. 3 and 13 are not consecutive primes.
Finally, the nomenclature for the NUMBER OF PRIMES LESS THAN OR EQUAL TO A CERTAIN NUMBER is π(n).
Here, the symbol pi is used to mean how many prime numbers are less than or equal to the number n in the parenthases. For example, π(5)= 3 because there are three prime numbers 2, 3 and 5 before and including 5. One of the fun challenges is π(100) which is 25 since there are 25 prime numbers among the first 100 whole numbers.
Incidentally, if your child does find the elusive pattern in the primes, I will accompany them and their family to Sweden or Norway to accept the prize. They do not have to share their prize. If I get there first, I will share my prize money with my current Mathletes ($1,000,000 divided by 170 is roughly $5,800 each). Yes, I am really still searching for this pattern.
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| Attachment | Size |
|---|---|
| Primes_first_170_from_1-1000.pdf | 357.54 KB |
| Prime_Activities.pdf | 42.78 KB |
| Primes_Activity_Answer_Key.pdf | 383.56 KB |
