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Topic: Palindromic Prime with an even number of digits (Read 2140 times) |
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mfirmata
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Palindromic Prime with an even number of digits
« on: Mar 15th, 2005, 2:41pm » |
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In a Clifford Pickover tear-a-day he asks, "What is the only palindromic prime number with an even number of digits?"
His answer is 11. 11 is certainly an example of a palindromic prime with an even number of digits, but how can he be sure it's the only one? What if there exists a prime number with, say, 100 billion and 2 digits in it that is also palindromic. Is there a proof for what he states?
At best, I'm a recreational mathematician, so this type of thing is out of league (not understanding a proof, just attempting one.)
I tried to e-mail Clifford Pickover (I've found other problems he posed whose answers were either wrong or else the problem was poorly stated) and it bounced back. I guess the story could be that he simply didn't apend, "under 1,000" to his question or something careless like that.
I apologize if this is in the wrong forum. Thanks
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Grimbal
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Re: Palindromic Prime with an even number of digit
« Reply #1 on: Mar 15th, 2005, 2:52pm » |
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I think it is true.
11 is a multiple of 11,
1001 is a multiple of 11,
100001 is a multiple of 11,
etc.
So, for instance, 314159951413 is a multiple of 11
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Icarus
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Re: Palindromic Prime with an even number of digit
« Reply #2 on: Mar 15th, 2005, 5:35pm » |
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Grimbal is correct. (xn + 1) = (x + 1)(xn-1 - xn-2 + ... - x + 1) when n is odd. In particular, when x =10 we get that 11 divides 10n + 1 for all odd n.
Since even digit palindromic numbers are all of the form [sum] ai(102i-1 + 1), where the ai are the digits of the palindrome, they are all divisible by 11.
So the only one that can be prime is 11 itself.
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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markr
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Re: Palindromic Prime with an even number of digit
« Reply #3 on: Mar 15th, 2005, 9:58pm » |
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A test for divisibility by 11 is to add the digits of a number (alternating the sign of each digit as you go). If the result is a multiple of 11, then so is the original number. In a palindrome of even length, the sum would be zero.
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