# More primes and squares, in a summation triangle

Place a different prime number or perfect square in each of the twenty-one disks that make up the triangle below, so that the number in any disk that lies on two others is precisely the sum of the numbers in those disks. Do so in such a way that the number in the apex is as small as possible.

Source: Based on a puzzle in here

A simple upper bound:

In barycentric coordinates: $$x_{abc} = 9^a 16^b 25^c$$. This gives a maximum at the tip $$x_{005} = 9765625$$.

And a brute-forced computer solution, barring coding mistakes this should be minimal:

                               147456
63487*    83969*
33211*    30276     53693*
17424     15787*    14489*    39204
10853*     6571*     9216      5273*    33931*
10529*     324       6247*     2969*     2304     31627*


The starred numbers are prime.

• Beautiful upper bound! Jan 10, 2021 at 14:33