Sequences of composite numbers: A contest
Procedure.
Choose two positive whole numbers x and y such that x < y, and y is prime. Construct the sequence x+y, x+2*y, x+3*y, until you again reach a prime number.
Contest.
You have to start with a number y < y0. Try to find a y that gives the longest sequence. If there is a draw, the smallest x+y wins.
Remarks.
In early grades use a small y0, for example, 100. In higher grades use y0 = 10,000 or more.
To generate your sequence, use a TI-108 calculator. The keystrokes are
[x][+][y][=][=] ... .
Checking whether a number is prime or composite is very difficult for large numbers. For this purpose students should use a TI-34 II or TI-83.
The procedure described above always terminates. This follows from a theorem proved by Johann Peter Gustav Lejeune Dirichlet (1805-1859) which says that if x and y are relatively prime, then the sequence x+y, x+2*y, x+3*y, ..., contains infinitely many prime numbers.
Examples.
|
x: |
y: (prime) |
x+y |
x+2y |
x+3y |
x+4y |
x+5y |
x+6y |
x+7y |
|
1 |
3 |
4 |
7 (prime) |
|
|
|
|
|
|
1 |
7 |
8 |
15 |
22 |
29 (prime) |
|
|
|
|
5 |
11 |
16 |
27 |
38 |
49 |
60 |
71 (prime) |
|
|
12 |
13 |
25 |
38 |
51 |
64 |
77 |
90 |
103 (prime) |