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.

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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:

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.


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