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Message: <blockquote data-quote="Novindu" data-source="post: 17393" data-attributes="member: 718">[From Patrick Capelle] 666 is the sum of the squares of two consecutive triangular numbers:666 = 15^2 + 21^2 which can also be elegantly written asT(6·6) = T(5)2 + T(6)2.But also note that T(5) + T(6) = T(8).  Indeed, 666 is the smallest triangular number of the form a^2 + b^2 with a+b also triangular.The doubly-triangular numbers are those numbers of the form T(T<img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="smilie smilie--sprite smilie--sprite23" alt="(n)" title="Thumbs down    (n)" loading="lazy" data-shortname="(n)" />), where T<img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="smilie smilie--sprite smilie--sprite23" alt="(n)" title="Thumbs down    (n)" loading="lazy" data-shortname="(n)" /> are the triangular numbers defined in the previous item.  The sequence of doubly-triangular numbers begins1, 6, 21, 55, 120, 321, 406, 666, 1035so we see that 666 is the eighth doubly-triangular number (i.e., T(T(8)) = 666).  The nth doubly-triangular number is, among other things, the number of ways to paint the vertices of a square using a set of n colors, where the colors are distinct but rotations and reflections of a given colored square are considered the same.  So there are 666 distinct ways of painting the vertices of a square with a set of eight colors.[from Monte Zerger]  6 (= T(3)), 66 (= T(11)), and 666 (= T(36)) are all triangular numbers in base 10.  These three numbers are also triangular in two other bases: 49 and 2040:(6)49 = 6 = T(3)(66)49 = 300 = T(24)(666)49 = 14706 = T(171)(6)2040 = 6 = T(3)(66)2040 = 12246 = T(1564)(666)2040 = 24981846 = T(7068)[from Monte Zerger]  666^6 = 87266061345623616, which contains 6 6's.  In addition, the digits of 666^6 can be split into two sets in two different ways, both of which sum up to the same value, 36 (= 6 x 6).The first eight and last nine digits both sum to 36:8 + 7 + 2 + 6 + 6 + 0 + 6 + 1 = 6 x 6 = 3 + 4 + 5 + 6 + 2 + 3 + 6 + 1 + 6while the 6's and non-6's also add up to 36:6 + 6 + 6 + 6 + 6 + 6 = 6 x 6 = 8 + 7 + 2 + 0 + 1 + 3 + 4 + 5 + 2 + 3 + 1Finally, note that 666^6 is almost pandigital - the only digit it's missing is an upside-down 6 (i.e., 9).A polygonal number is a positive integer of the form P(k,n) = n((k - 2)n + 4 - k)/2 where k is the 'order' of the polygonal number (k=3 gives the triangular numbers, k=4 the squares, k=5 the pentagonal numbers, etc.), and n is its index. A repdigit polygonal number is a polygonal number that also happens to be a repdigit. Finally, define the wickedness of a polygonal number as n/k. Now, an amazing fact:666 is conjectured to be the most wicked repdigit polygonal number.Since 666 = P(3,36), its wickedness value is n/k = 12. I recently showed by computer calculation that there are no counterexamples to this conjecture less than 10^50. See my paper here for more details. It seems quite certain that this is true but so far no one has proved it.Whilst on the subject of polygonal numbers, we can find among them some rather beastly configurations. One of the more striking is the following:If one arranges a group of people in a solid 3010529326318802-sided polygon with 666 people on each side, there will be a total of 666666666666666666666 persons in all.Or, more simply, P(3010529326318802, 666) = 666666666666666666666. See the paper link in the previous item for more like this.Define PI(n,d) as the d consecutive decimal digits of the number π (3.14159265358979...) starting at the nth digit after the decimal point. Then we can make the following pretty statement: PI(666, 3) = 7·7·7 (since the digits at that position are "343", or 7 cubed)as well as the following one, which contains nothing but 6's and 3's (and two 666's): PI(666 · 3.663663663..., 3) = 666.Inserting zeros between the sixes in 666 gives the number 60606, which has a few interesting properties of its own:60606 = 7 x 13 x 666 = 91 x 666 = T(13) x T(36) - i.e., 60606 is the product of two triangular numbers.60606 = 7 x 37 x (13 x 18), which is interesting in that Rev 13:18 is the place where 666 is mentioned.60606 = P(7,156) - i.e., 60606 is a 7-gonal number. (Note that this can be written entirely using the evocative numbers 6, 7, and 13, by saying 60606 = P(7, (6+6)·13)). In addition we can make a statement using only 7's: 60606 is the 7th palindromic 7-gonal number. 60606 has exactly 6 prime factors.60606+1 is a prime number. Not only that, but it's a prime (p) for which the period length of the decimal expansion of its reciprocal (1/p) attains the maximum possible value of p-1. In other words:1/(60606 + 1) has period length 60606.60606 is, just like 666, the sum of two consecutive palindromic primes (both of which contain the evil eyes!):60606 = 30203 + 30403.[Thanks to G. L. Honaker, Jr., Jud McCranie, Monte Zerger, and Patrick De Geest for these.]</blockquote>

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