## The ‘Happy Holidays’ number

December 23rd, 2015 | Categories: general math, just for fun, Maple, math software | Tags:

Some numbers have something to say. Take the following, rather huge number, for example:

185325291040682644803531312384041336595151018761127807725763308064246070395230764956468856341399670487
514610052487586323067575687914642829757636555138456145938430191876551756992329818006401775522301219016
237245425891544032218544390861818271526845858747648909382915665997160517028671058273052955697138350617
856171748990490346558484883522495310587304606877332488244886849690319641412147118669050542398759303832
627672479768452329971883073420877438596419179762421854464516060347269129680634374662501202129049727949
71185874579656679344857677824

This number wants to tell you ‘Happy Holidays’, it just needs a little code to help it out.  In Maple, this code is:

```n := 18532529104068264480353131238404133659515101876112780772576330806424607039523076495646885634139967048751461005248758632306757568791464282975763655513845614593843019187655175699232981800640177552230121901623724542589154403221854439086181827152684585874764890938291566599716051702867105827305295569713835061785617174899049034655848488352249531058730460687733248824488684969031964141214711866905054239875930383262767247976845232997188307342087743859641917976242185446451606034726912968063437466250120212904972794971185874579656679344857677824:
modnew := proc (x, y) options operator, arrow; x-y*floor(x/y) end proc:
tupper := piecewise(1/2 < floor(modnew(floor((1/17)*y)*2^(-17*floor(x)-modnew(floor(y), 17)), 2)), 0, 1):
points := [seq([seq(tupper(x, y), y = n+16 .. n, -1)], x = 105 .. 0, -1)]:
plots:-listdensityplot(points, scaling = constrained, view = [0 .. 106, 0 .. 17], style = patchnogrid, size = [800, 800]);
```

The result is the following plot

Thanks to Samir for this one!

The mathematics is based on a generalisation of Tupper’s self-referential formula.

There’s more than one way to send a message with an equation, however. Here’s an image of one I discovered a few years ago — The equation that says Hi