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

 
 
Octagonal numbers are of the form    O_{n} \; = \; n \, (3 \, n - 2)

 

We can write

10^{n} \; x \; + \; (x + 1) \; = \; y \, (3 \, y - 2)

Or

(10^{n} + 1) \, x \; + \; 1 \; = \; y \, (3 \, y - 2)

the integer length of   x   is   n

 
Here are the first few solutions
 

n   =   1,   2,   5,   10 :     no solutions

n = 3

100 \; || \; 101 \; = \; 100101 \; = \; O_{183}
704 \; || \; 705 \; = \; 704705 \; = \; O_{485}

n = 4

8132 \; || \; 8133 \; = \; 81328133 \; = \; O_{5207}

n = 6

142340 \; || \; 142341 \; = \; 142340142341 \; = \; O_{217823}
604384 \; || \; 604385 \; = \; 604384604385 \; = \; O_{448845}

n = 7

2754820 \; || \; 2754821 \; = \; 27548202754821 \; = \; O_{3030303}
3966944 \; || \; 3966945 \; = \; 39669443966945 \; = \; O_{3636365}

n = 8

29527104 \; || \; 29527105 \; = \; 2952710429527105 \; = \; O_{31372549}
37370244 \; || \; 37370245 \; = \; 3737024437370245 \; = \; O_{35294119}

n = 9

123636159 \; || \; 123636160 \; = \; 123636159123636160 \; = \; O_{203007520}
147556740 \; || \; 147556741 \; = \; 147556740147556741 \; = \; O_{221778223}
238451204 \; || \; 238451205 \; = \; 238451204238451205 \; = \; O_{281928599}
267667399 \; || \; 267667400 \; = \; 267667399267667400 \; = \; O_{298701300}
284250207 \; || \; 284250208 \; = \; 284250207284250208 \; = \; O_{307814992}
386323575 \; || \; 386323576 \; = \; 386323575386323576 \; = \; O_{358851676}
406195535 \; || \; 406195536 \; = \; 406195535406195536 \; = \; O_{367965368}
444070144 \; || \; 444070145 \; = \; 444070144444070145 \; = \; O_{384738069}
593777184 \; || \; 593777185 \; = \; 593777184593777185 \; = \; O_{444888445}
644939415 \; || \; 644939416 \; = \; 644939415644939416 \; = \; O_{463659148}
733607055 \; || \; 733607056 \; = \; 733607055733607056 \; = \; O_{494505496}
876687424 \; || \; 876687425 \; = \; 876687424876687425 \; = \; O_{540582225}
922929407 \; || \; 922929408 \; = \; 922929407922929408 \; = \; O_{554655872}
979591840 \; || \; 979591841 \; = \; 979591840979591841 \; = \; O_{571428573}

 
 

O_{485} \; = \; 704705 \; = \; 704 \; || \; 705
O_{448845} \; = \; 604384604385 \; = \; 604384 \; || \; 604385
O_{444888445} \; = \; 593777184593777185 \; = \; 593777184 \; || \; 593777185
O_{444488884445} \; = \; 592711105184592711105185 \; = \; 592711105184 \; || \; 592711105185

the pattern continues

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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