Problems & Puzzles:
Puzzles
Puzzle 132.
Pascal
Primes
For
sure you know the Pascal triangle:
| Row |
Pascal triangle |
| 1 |
1 |
| 2 |
1
1 |
| 3 |
1 2 1 |
| 4 |
1 3 3 1 |
| 5 |
1 4 6 4 1 |
| 6 |
1 5 10 10 5 1 |
| 7 |
1 6 15 20 15 6 1 |
| 8 |
1 7 21 35 35 21 7 1 |
| 9 |
1
8 28 56 70 56 28 8 1 |
| etc |
etc |
Concatenating
the numbers in each row, the following rows are primes: 2, 9, 30, ?
Questions:
a)
Find the fourth prime-row.
b)
The 9th row reversed is prime also. What other rows reversed are primes?
Solution
Paul Jobling sent the following (10/4/01) for our question b)
"You ask about reversed rows being prime.
row 2 = 11
row 7 = 1651025161
row 9 = 18826507658281
row 10 = 196348621621486391
row 11 = 1015402101225201202154011
row 33=12369406940695367310229160965856330038150100884/
082042215460844209210482975220063737430065341740272275/
650930801060272275650065341740063737430482975220844209/
210422154600884082003815016585633291609673102069530694694231
Titanix was used to prove these prime".
(Note: this doesn't mean that these primes are reversible)
***
J. K. Andersen wrote (Jun 2003):
PrimeForm/GW found only the already known primes in the first 696 rows
and reversed rows. The search stopped before row 697 which has 104568
digits if anyone feels lucky.
***
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