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Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number ...
Centered hexagonal numbers appearing in the Catan board game: 19 land tiles, 37 total tiles. In mathematics and combinatorics, a centered hexagonal number, or centered hexagon number, [1] [2] is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice.
visual proof centered hexagonal numbers sum: Image title: Proof without words of the sum of the first n hex numbers by arranging n³ balls in a cube and viewing along a body diagonal, by CMG Lee. Colour denotes cube layer and line style denotes hex number. Width: 100%: Height: 100%
Date/Time Thumbnail Dimensions User Comment; current: 21:27, 13 September 2023: 512 × 640 (6 KB): Cmglee: Form triangular numbers more intuitively: 23:44, 23 August 2023
Figurate numbers were a concern of the Pythagorean worldview. It was well understood that some numbers could have many figurations, e.g. 36 is a both a square and a triangle and also various rectangles. The modern study of figurate numbers goes back to Pierre de Fermat, specifically the Fermat polygonal number theorem.
This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal. As of July 2023 [update] the largest known has 3,153,105 digits with y = 3 3304301 − 1 {\displaystyle y=3^{3304301}-1} , [ 2 ] found by R.Propper and S.Batalov.
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