Prime Triangle

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Published on 2011. február 14. hétfő, 07:57

I studied the primes in my high school period and this one was a still memorable observation of mine.

The rows of the difference-triangle are enumerated starting with row $n = 0$ at the top. The entries in each row are numbered from the left beginning with $k = 0$ and are staggered relative to the numbers in the adjacent rows. A simple construction of the difference-triangle proceeds in the following manner. On row 0, write the elements of the sequence. Then, to construct the elements of the following rows, grab the number directly above and to the left and the number directly above and to the right and take the absolut value of their difference to find the new value. If either the number to the right or left is not present, leave the place empty.

Considering the difference-triangle of an arithmetic sequence is not exciting because all the elements of the second line is zero. Considering the difference-triangle of a geometric sequence of powers of a natural number $q$ is more complicated, but we can calculate the $k$-th element $e_{n,k}$ of the $n$-th row as $$ e_{n,k} =\sum_{i=k}^{k+n}(-1)^{k+n-i}q^{i}{n\choose i-k} =\sum_{j=0}^{n}(-1)^{n-j}q^{j+k}{n\choose j} =q^k(q-1)^{n}. $$ This triangle is shown for $q=3$ in the table below:

The beginning of the difference-triangle of the primes is shown in the table below:

What is interesting in this is that all the elements of the frist left diagonal is 1! I made a computer program in java to justify this observation for much longer (in fact it is done at the moment for the first 7048220 primes -- that last prime is 123847351--!) and reestablished my conjecture formulated first in 1978:

Every elements of the first left-diagonal of the table made from primes are 1!

Of course, other sequences can be as well examined in the same facility. For example, the sequence of powers $2^n$ of two is such that every elements of its first left-diagonal are 1. This is shown in the table below:

It turned out with the help of "The On-Line Encyclopedia of Integer Sequences", that this observation was already taken by Proth in 1878, and he had given a faulty proof also, and then in 1958 Norman O. Gilbreath found again this table. It is still not proven, nevertheless in 1993 Andrew Odlyzko had checked it using the primes up to 10,000,000,000,000 (that is 346,065,536,839 rows)!

Related pages

Search on The On-Line Encyclopedia of Integer Sequences
Find on The Prime Pages Glossary
Andrew Odlyzko: Iterated absolute values of differences of consecutive primes, Math. Comp., 61 (1993), 373--380.