Puzzle about prime numbers?

Question

Take prime numbers: 2,3,5,7,11,13,... and write the difference between any consecutive numbers: 1,2,2,4,2,...; then do the same with this new sequence: 1,0,2,2,...; and over and over...
Can you prove that the first number is always 1 (except the original sequence of primes of course)?

Clearly by my examples it is understood that I intended "distance" when I said "difference" (that is, absolute value you know)

To neshal.chanderman.ramanujan: how do you prove that "the value of the first cell is in absolute value less than 2" in every row of differences? Please give details.

To Joe H: where did that 9 come from?! I bet it was a mayonnaise stain on your paper.

Answer 1

It's quite trivial to resolve your puzzle with the bean method. Try at home, it's even funny.

Answer 2

My sneaking suspicion is that this is one of those unproven conjectures about prime numbers that is almost certainly true but extremely difficult to prove, kind of like the conjecture that any even number can be expressed as the sum of two primes. If this really is provable, I'd sure be interested in seeing the logic.

As for the likelihood that this is almost certainly true, try the test on the primes under 100, and you'll see why there isn't much chance of the statement being false. First, it only takes three iterations before the great majority of the differences (at least for the low numbers) are 0 or 2, and you can't get the first number to be something other than 1 if the second number is 0 or 2. You need a difference of 4 or more to eventually end up in the second position, but the 4s in the lower positions after the second iteration match up with 2s, effectively knocking them out, and the 4+ differences still around after the third iteration are way down the line -- seems hopeless to expect any of them not to run into enough 2s to turn them into 2s as well.

Answer 3

I'm not sure if this will help, but...

If you take the differences (m-n) between adjacent numbers in your series, without taking absolute values ABS(m-n), then the first number in the series is given by a binomial expression of the prime numbers up to (j+1)-th number for the j-th series of differences. This would give a prime number generator if another expression can be found for the first term which, in this case, fluctuates between rapidly diverging positive and negative values.

In your case, where all differences are given positive values ABS(m-n), then I don't think it can be formed into such a straightforward expression when j > 2 and it is necessary to number-crunch on a computer. To get the (j+1)-th prime, the computer needs to calculate 2^j ABS( ) terms. Hence, assuming the SMVC conjecture that the first term is always 1, to get the 1001-th prime requires 2^1000 terms plus a decision as to whether the first difference is +1 or -1.

Perhaps a mathematician will prove me wrong, however.

Answer 4

the simplest way to see this is to create a listing of the first ten numbers in excel and create about 5 rows of differences.

The solution follows on noting

-- that the second and subsequent columns can only be even.

-- the value of the first cell is in absolute value less than 2.

It is thus an odd number less than 2 = 1 QED

both steps are easy to prove, drop a line if you want further help on either

Answer 5

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Answer 6

the proof ofthat i donot know.i have t work on that.