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Determine the least possible value of the largest term in an arithmetic progression of seven distinct primes.
Source: BMO1 2004 Q4
To avoid suspense, I'll tell you now the answer is 907.
Did you try this problem and get 1307 as your answer?
If so, you probably reasoned that the sequence is a, a+d, a+2d,..., a+6d. In order for all of these numbers to be prime, the common difference, d, must be a multiple of 2, 3, 5, and 7 because otherwise one of these numbers would be a multiple of 7. So d is a multiple of 210. Let's try d=210. Trying prime numbers in turn, you probably saw a=47 works. 47, 257, 467, 677, 887, 1097, and 1307 are all prime.
Now, what you failed to consider: a can be a multiple of 7 if (and only if) it is, in fact 7. In that case, d doesn't need to be a multiple of 7, so d can be 30 or a multiple of 30. So let's try various multiples of 30 for d, and see if we get 7 primes:
if d=30 then the sequence is 7, 37, 67, 97, 127, 157, 187, but 187 is
divisible by 11;
if d=60 then the sequence is 7, 67, 127, 187, 247, 307, 367, but 187 is
divisible by 11;
if d=90 then the sequence is 7, 97, 187, 277, 367, 457, 547, but 187 is
divisible by 11;
if d=120 then the sequence is 7, 127, 247, 367, 487, 607, 727, but 247 is
divisible by 13;
if d=150 then the sequence is 7, 157, 307, 457, 607, 757, 907 -- all prime!
The least possible value of the largest term in an arithmetic progression of n distinct primes (OEIS A005115) is given by the following table:
| n | value | sequence |
| 1 | 2 | 2 |
| 2 | 3 | 2, 3 |
| 3 | 7 | 3, 5, 7 |
| 4 | 23 | 5, 11, 17, 23 |
| 5 | 29 | 5, 11, 17, 23, 29 |
| 6 | 157 | 7, 37, 67, 97, 127, 157 |
| 7 | 907 | 7, 157, 307, 457, 607, 757, 907 |
| 8 | 1669 | 199, 409, 619, 829, 1039, 1249, 1459, 1669 |
| 9 | 1879 | 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879 |
| 10 | 2089 | 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 |
| 11 | 249037 | 110437, 124297, 138157, 152017, 165877, 179737, 193597, 207457, 221317, 235177, 249037 |
| 12 | 262897 | 110437, 124297, 138157, 152017, 165877, 179737, 193597, 207457, 221317, 235177, 249037, 262897 |
| 13 | 725663 | 4943, 65003, 125063, 185123, 245183, 305243, 365303, 425363, 485423, 545483, 605543, 665603, 725663 |
| 14 | 36850999 | 31385539, 31805959, 32226379, 32646799, 33067219, 33487639, 33908059, 34328479, 34748899, 35169319, 35589739, 36010159, 36430579, 36850999 |
OEIS A005115
(none)
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