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

 

Oblong numbers are of the form    O_{n} \; = \; n \, (n + 1)

 

To find solutions to

x \; || \; (x + 1) \; = \; y \,(y + 1)

 

We can write

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

Or

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

the integer length of   x   is   n

 
Here are the first few solutions
 

n = 1

1 \; || \; 2 \; = \; 12 \; = \; O_3
5 \; || \; 6 \; = \; 56 \; = \; O_7

1 \; || \; 2 \; = \; 3 \,(3 + 1)
5 \; || \; 6 \; = \; 7 \,(7 + 1)

n = 2

61 \; || \; 62 \; = \; 6162 \; = \; O_{78}

61 \; || \; 62 \; = \; 78 \,(78 + 1)

 

n   =   3,   4 :     no solutions

 

n = 5

65479 \; || \; 65480 \; = \; 6547965480 \; = \; O_{80919}
84289 \; || \; 84290 \; = \; 8428984290 \; = \; O_{91809}

65479 \; || \; (65479 + 1) \; = \; 80919 \,(80919 + 1)
84289 \; || \; (84289 + 1) \; = \; 91809 \,(91809 + 1)

n = 6

106609 \; || \; 106610 \; = \; 106609106610 \; = \; O_{326510}
225649 \; || \; 225650 \; = \; 225649225650 \; = \; O_{475025}
275599 \; || \; 275600 \; = \; 275599275600 \; = \; O_{524975}
453589 \; || \; 453590 \; = \; 453589453590 \; = \; O_{673490}

106609 \; || \; (106609 + 1) \; = \; 326510 \,(326510 + 1)
225649 \; || \; (225649 + 1) \; = \; 475025 \,(475025 + 1)
275599 \; || \; (275599 + 1) \; = \; 524975 \,(524975 + 1)
453589 \; || \; (453589 + 1) \; = \; 673490 \,(673490 + 1)

n = 7

1869505 \; || \; 1869506 \; = \; 18695051869506 \; = \; O_{4323777}
2272555 \; || \; 2272556 \; = \; 22725552272556 \; = \; O_{4767132}
2738291 \; || \; 2738292 \; = \; 27382912738292 \; = \; O_{5232868}
3221951 \; || \; 3221952 \; = \; 32219513221952 \; = \; O_{5676223}

1869505 \; || \; (1869505 + 1) \; = \; 4323777 \,(4323777 + 1)
2272555 \; || \; (2272555 + 1) \; = \; 4767132 \,(4767132 + 1)
2738291 \; || \; (2738291 + 1) \; = \; 5232868 \,(5232868 + 1)
3221951 \; || \; (3221951 + 1) \; = \; 5676223 \,(5676223 + 1)

 

n   =   8,   9 :     no solutions

 

n = 10

1667833021 \; || \; 1667833022 \; = \; 16678330211667833022 \; = \; O_{4083911141}
2475062749 \; || \; 2475062750 \; = \; 24750627492475062750 \; = \; O_{4975000250}
2525062249 \; || \; 2525062250 \; = \; 25250622492525062250 \; = \; O_{5024999750}
3500010739 \; || \; 3500010740 \; = \; 35000107393500010740 \; = \; O_{5916088859}
9032526511 \; || \; 9032526512 \; = \; 90325265119032526512 \; = \; O_{9503960496}
9225507211 \; || \; 9225507212 \; = \; 92255072119225507212 \; = \; O_{9604950396}

1667833021 \; || \; (1667833021 + 1) \; = \; 4083911141 \,(4083911141 + 1)
2475062749 \; || \; (2475062749 + 1) \; = \; 4975000250 \,(4975000250 + 1)
2525062249 \; || \; (2525062249 + 1) \; = \; 5024999750 \,(5024999750 + 1)
3500010739 \; || \; (3500010739 + 1) \; = \; 5916088859 \,(5916088859 + 1)
9032526511 \; || \; (9032526511 + 1) \; = \; 9503960496 \,(9503960496 + 1)
9225507211 \; || \; (9225507211 + 1) \; = \; 9604950396 \,(9604950396 + 1)

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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