I'm sure everyone's aware of the concept of iteratively taking differences
between consecutive terms in sequences of integers. (Ouch, I think I've just
made it sound more complicated than it is!) For a finite seqence, each line
of differences is one shorter than the one it's below, so that a triangle is
formed.
e.g.
1 4 9 16
3 5 7
2 2
0

Well, as this is the Primes List, let's throw primes into the mix - I only
want to see primes in the triangle. Of course, differences between odd
primes will be even, so I'm going to throw a factor of two into the mix too,
divide the differences between the above terms by two to form subsequent
rows. Note, explicitly, zero and one are _not_ primes. I'm also only
interested in positive terms, which means that all of the rows must be
strictly increasing.

E.g.
3 7
2
and
5 19 41
7 11
2
are both valid triangles.

But neither of
3 5 or 2 2
1 0
are valid.

For each triangle size, what's the smallest total sum of all terms?
e.g. the first valid one above has a term sum of 12, and the second one has
a term sum of 85.
Can you find a 3-triangle with a sum less than 71?

Variations:
a) What happens if I forbid any prime from being repeated in the triangle?
Does the minimum 3-triangle have a term sum of 85, as seen above?

b) What happens if I now permit negative primes into the triangle formed
by taking (half) the signed difference in the usual right-left order (as all
the examples above). This time I'm interested in minimising the sum of the
absolute values in the tables.
e.g.
-13 -7 7
3 7
2 abs. sum = 39
Why do I want to forbid repetitions (i.e. include variation (a) too)
when I adopt this variation? When I do forbid repetitions, what's
the minimum absolute sum?

Is there a general rule for constructing these minimally-summing triangles?
(It's not immedately clear from the 2- and 3-sized solutions I have.)
If not, then what's the largest triangle that someone is prepared to assert
cannot be bettered?

As you can tell, I have essential chores to do, and am doing anything to
avoid doing them! :-)

Phil


=====
First rule of Factor Club - you do not talk about Factor Club.
Second rule of Factor Club - you DO NOT talk about Factor Club.
Third rule of Factor Club - when the cofactor is prime, or you've trial-
divided up to the square root of the number, the factoring is over.

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