When the Concatenation of n || (n+1) is Hexagonal

 
 
Hexagonal numbers are of the form    n \, (2 \, n - 1)

 

We can write

10^{n} \; x \; + \; (x + 1) \; = \; y \, (2 \, y - 1)

Or

(10^{n} + 1) \, x \; + \; 1 \; = \; y \, (2 \, y - 1)

the integer length of   x   is   n

 
Here are the first few solutions
 

n = 1

4 \, || \, 5 \, = \, 45 \, = \, H_5

n = 2

49 \, || \, 50 \, = \, 4950 \, = \, H_{50}

n = 3

369 \, || \, 370 \, = \, 369370 \, = \, H_{430}
415 \, || \, 416 \, = \, 415416 \, = \, H_{456}
499 \, || \, 500 \, = \, 499500 \, = \, H_{500}
760 \, || \, 761 \, = \, 760761 \, = \, H_{617}

n = 4

4999 \, || \, 5000 \, = \, 49995000 \, = \, H_{5000}
8827 \, || \, 8828 \, = \, 88278828 \, = \, H_{6644}

n = 5

10330 \, || \, 10331 \, = \, 1033010331 \, = \, H_{22727}
14877 \, || \, 14878 \, = \, 1487714878 \, = \, H_{27274}
49999 \, || \, 50000 \, = \, 4999950000 \, = \, H_{50000}

n = 6

499999 \, || \, 500000 \, = \, 499999500000 \, = \, H_{500000}
490150 \, || \, 490151 \, = \, 490150490151 \, = \, H_{495051}

n = 7

4999999 \, || \, 5000000 \, = \, 49999995000000 \, = \, H_{5000000}
6983470 \, || \, 6983471 \, = \, 69834706983471 \, = \, H_{5909091}

n = 8

49999999 \, || \, 50000000 = 4999999950000000 \, = \, H_{50000000}
83737027 \, || \, 83737028 = 8373702783737028 \, = \, H_{64705884}

n = 9

167590027 \, || \, 167590028 \; = \; 167590027167590028 \; = \; H_{289473684}
215738734 \, || \, 215738735 \; = \; 215738734215738735 \; = \; H_{328434723}
249759739 \, || \, 249759740 \; = \; 249759739249759740 \; = \; H_{353383460}
254388969 \, || \, 254388970 \; = \; 254388969254388970 \; = \; H_{356643358}
255102040 \, || \, 255102041 \; = \; 255102040255102041 \; = \; H_{357142857}
444232045 \, || \, 444232046 \; = \; 444232045444232046 \; = \; H_{471291866}
499001500 \, || \, 499001501 \; = \; 499001500499001501 \; = \; H_{499500501}
499999999 \, || \, 500000000 \; = \; 499999999500000000 \; = \; H_{500000000}
579881659 \, || \, 579881660 \; = \; 579881659579881660 \; = \; H_{538461540}
580958002 \, || \, 580958003 \; = \; 580958002580958003 \; = \; H_{538961039}
643362877 \, || \, 643362878 \; = \; 643362877643362878 \; = \; H_{567169674}
739705014 \, || \, 739705015 \; = \; 739705014739705015 \; = \; H_{608155003}
928390294 \, || \, 928390295 \; = \; 928390294928390295 \; = \; H_{681318683}
929752065 \, || \, 929752066 \; = \; 929752065929752066 \; = \; H_{681818182}

n = 10

1066727340 \, || \, 1066727341 \; = \; 10667273401066727341 \; = \; H_{2309466757}
1447793827 \, || \, 1447793828 \; = \; 14477938271447793828 \; = \; H_{2690533244}
1764532890 \, || \, 1764532891 \; = \; 17645328901764532891 \; = \; H_{2970297031}
4999999999 \, || \, 5000000000 \; = \; 49999999995000000000 \; = \; H_{5000000000}
6408999877 \, || \, 6408999878 \; = \; 64089998776408999878 \; = \; H_{5660830274}

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , . Bookmark the permalink.

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