## When the Concatenation of n || (n+1) is triangular

Solving

$x \; || \; (x + 1)$   is a triangular number

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

Here are the first few solutions:

$4 \; || \; 5 \; = \; 45 \; = \; T_9$
$7 \; || \; 8 \; = \; 78 \; = \; T_{12}$

$52 \; || \; 53 \; = \; 5253 \; = \; T_{102}$
$49 \; || \; 50 \; = \; 4950 \; = \; T_{99}$

$295 \; || \; 296 \; = \; 295296 \; = \; T_{768}$
$369 \; || \; 370 \; = \; 369370 \; = \; T_{859}$
$415 \; || \; 416 \; = \; 415416 \; = \; T_{911}$
$499 \; || \; 500 \; = \; 499500 \; = \; T_{999}$
$502 \; || \; 503 \; = \; 502503 \; = \; T_{1002}$
$594 \; || \; 595 \; = \; 594595 \; = \; T_{1090}$
$652 \; || \; 653 \; = \; 652653 \; = \; T_{1142}$
$760 \; || \; 761 \; = \; 760761 \; = \; T_{1233}$

$2254 \; || \; 2255 \; = \; 22542255 \; = \; T_{6714}$
$4999 \; || \; 5000 \; = \; 49995000 \; = \; T_{9999}$
$5002 \; || \; 5003 \; = \; 50025003 \; = \; T_{10002}$
$8827 \; || \; 8828 \; = \; 88278828 \; = \; T_{13287}$

$10330 \; || \; 10331 \; = \; 1033010331 \; = \; T_{45453}$
$14877 \; || \; 14878 \; = \; 1487714878 \; = \; T_{54547}$
$49999 \; || \; 50000 \; = \; 4999950000 \; = \; T_{99999}$
$50002 \; || \; 50003 \; = \; 5000250003 \; = \; T_{100002}$

$490150 \; || \; 490151 \; = \; 490150490151 \; = \; T_{990101}$
$499999 \; || \; 500000 \; = \; 499999500000 \; = \; T_{999999}$
$500002 \; || \; 500003 \; = \; 500002500003 \; = \; T_{1000002}$
$509949 \; || \; 509950 \; = \; 509949509950 \; = \; T_{1009900}$

$3347109 \; || \; 3347110 \; = \; 33471093347110 \; = \; T_{8181820}$
$4999999 \; || \; 5000000 \; = \; 49999995000000 \; = \; T_{9999999}$
$5000002 \; || \; 5000003 \; = \; 50000025000003 \; = \; T_{10000002}$
$6983470 \; || \; 6983471 \; = \; 69834706983471 \; = \; T_{11818181}$

$24913494 \; || \; 24913495 \; = \; 2491349424913495 \; = \; T_{70588234}$
$49999999 \; || \; 50000000 \; = \; 4999999950000000 \; = \; T_{99999999}$
$50000002 \; || \; 50000003 \; = \; 5000000250000003 \; = \; T_{100000002}$
$83737027 \; || \; 83737028 \; = \; 8373702783737028 \; = \; T_{129411767}$

$123989904 \; || \; 123989905 \; = \; 123989904123989905 \; = \; T_{497975710}$
$126014194 \; || \; 126014195 \; = \; 126014194126014195 \; = \; T_{502024290}$
$167590027 \; || \; 167590028 \; = \; 167590027167590028 \; = \; T_{578947367}$
$168168895 \; || \; 168168896 \; = \; 168168895168168896 \; = \; T_{579946368}$
$202479340 \; || \; 202479341 \; = \; 202479340202479341 \; = \; T_{636363638}$
$203115565 \; || \; 203115566 \; = \; 203115565203115566 \; = \; T_{637362636}$
$215738734 \; || \; 215738735 \; = \; 215738734215738735 \; = \; T_{656869445}$
$254388969 \; || \; 254388970 \; = \; 254388969254388970 \; = \; T_{713286715}$
$255102040 \; || \; 255102041 \; = \; 255102040255102041 \; = \; T_{714285713}$
$307085005 \; || \; 307085006 \; = \; 307085005307085006 \; = \; T_{783689996}$
$374684184 \; || \; 374684185 \; = \; 374684184374684185 \; = \; T_{865660654}$
$425113849 \; || \; 425113850 \; = \; 425113849425113850 \; = \; T_{922077924}$
$426035502 \; || \; 426035503 \; = \; 426035502426035503 \; = \; T_{923076922}$
$444232045 \; || \; 444232046 \; = \; 444232045444232046 \; = \; T_{942583731}$
$499001500 \; || \; 499001501 \; = \; 499001500499001501 \; = \; T_{999001001}$
$499999999 \; || \; 500000000 \; = \; 499999999500000000 \; = \; T_{999999999}$
$500000002 \; || \; 500000003 \; = \; 500000002500000003 \; = \; T_{1000000002}$
$500999499 \; || \; 500999500 \; = \; 500999499500999500 \; = \; T_{1000999000}$
$559064584 \; || \; 559064585 \; = \; 559064584559064585 \; = \; T_{1057416270}$
$579881659 \; || \; 579881660 \; = \; 579881659579881660 \; = \; T_{1076923079}$
$580958002 \; || \; 580958003 \; = \; 580958002580958003 \; = \; T_{1077922077}$
$643362877 \; || \; 643362878 \; = \; 643362877643362878 \; = \; T_{1134339347}$
$739705014 \; || \; 739705015 \; = \; 739705014739705015 \; = \; T_{1216310005}$
$826530615 \; || \; 826530616 \; = \; 826530615826530616 \; = \; T_{1285714288}$
$827815540 \; || \; 827815541 \; = \; 827815540827815541 \; = \; T_{1286713286}$
$836225902 \; || \; 836225903 \; = \; 836225902836225903 \; = \; T_{1293233082}$
$901999845 \; || \; 901999846 \; = \; 901999845901999846 \; = \; T_{1343130556}$
$928390294 \; || \; 928390295 \; = \; 928390294928390295 \; = \; T_{1362637365}$
$929752065 \; || \; 929752066 \; = \; 929752065929752066 \; = \; T_{1363636363}$

math grad - Interest: Number theory
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### 4 Responses to When the Concatenation of n || (n+1) is triangular

1. paul says:

There is an unusual pattern amongst these numbers, there always seems to be many more solutions when the integer length of x is 9, in other words 18 digit concatenated numbers. for all types of number i.e. square, triangular, pentagonal, octagonal… … up to hexagonal type numbers. and the hexagonal numbers are the first that do not have a 20 digit solution.

Paul.

• benvitalis says:

Hexagonal numbers: n*(2*n-1)
when integer length of x is 9
(10^9 + 1)*x + 1 = y*(2*y-1)
when integer length of x is 10
(10^10 + 1)*x + 1 = y*(2*y-1)

lots of solutions

And (10^n + 1)*x, n ≥ 9 ( I have checked for n up to 20 and for Triangular,Pentagonal,Hexagonal,Heptagonal,Octagonal and Nonagonal

2. paul says:

Sorry I meant to say octagonal numbers, have no 20 digit solutions, (x=10). y(3y -2) type.
P.