## Pentagonal & Triangular numbers; P_x – T_y = 1

A number of the form   $T_n \; = \; n \,(n + 1) \,/ \,2$ is called a triangular number.

The n-th pentagonal number is given by the formula   $P_n \; = \; n \, (3 \, n-1) \,/ \,2$

Let’s find pentagonal numbers   $P_x$   and Triangular numbers   $T_y$   such that

$P_x \; - \; T_y \; = \; 1$

$x \, (3 \, x - 1)/2 \; - \; y \,(y + 1)/2 \; = \; 1$

$1/2 \; (3 \, x^2 \; - \; x \; - \; y^2 \; - \; y) \; = \; 1$

Here are the first few solutions:

$21 \; = \; T_6$
$22 \; = \; P_4$

$91 \; = \; T_{13}$
$92 \; = \; P_8$

$4186 \; = \; T_{91}$
$4187 \; = \; P_{53}$

$17766 \; = \; T_{188}$
$17767 \; = \; P_{109}$

$812175 \; = \; T_{1274}$
$812176 \; = \; P_{736}$

$3446625 \; = \; T_{2625}$
$3446626 \; = \; P_{1516}$

$157557876 \; = \; T_{17751}$
$157557877 \; = \; P_{10249}$

$668627596 \; = \; T_{36568}$
$668627597 \; = \; P_{21113}$

$30565415881 \; = \; T_{247246}$
$30565415882 \; = \; P_{142748}$

$129710307111 \; = \; T_{509333}$
$129710307112 \; = \; P_{294064}$

$5929533123150 \; = \; T_{3443699}$
$5929533123151 \; = \; P_{1988221}$

$25163130952050 \; = \; T_{7094100}$
$25163130952051 \; = \; P_{4095781}$

$1150298860475331 \; = \; T_{47964546}$
$1150298860475332 \; = \; P_{27692344}$

$4881517694390701 \; = \; T_{98808073}$
$4881517694390702 \; = \; P_{57046868}$

$223152049399091176 \; = \; T_{668059951}$
$223152049399091177 \; = \; P_{385704593}$

$946989269580844056 \; = \; T_{1376218928}$
$946989269580844057 \; = \; P_{794560369}$

$43290347284563212925 \; = \; T_{9304874774}$
$43290347284563212926 \; = \; P_{5372171956}$

$183711036780989356275 \; = \; T_{19168256925}$
$183711036780989356276 \; = \; P_{11066798296}$

$8398104221155864216386 \; = \; T_{129600186891}$
$8398104221155864216387 \; = \; P_{74824702789}$

$35638994146242354273406 \; = \; T_{266979378028}$
$35638994146242354273407 \; = \; P_{154140615773}$

$1629188928556953094766071 \; = \; T_{1805097741706}$
$1629188928556953094766072 \; = \; P_{1042173667088}$

$6913781153334235739684601 \; = \; T_{3718543035473}$
$6913781153334235739684602 \; = \; P_{2146901822524}$

$316054254035827744520401500 \; = \; T_{25141768196999}$
$316054254035827744520401501 \; = \; P_{14515606636441}$

$1341237904752695491144539300 \; = \; T_{51792623118600}$
$1341237904752695491144539301 \; = \; P_{29902484899561}$

$61312896094022025483863125041 \; = \; T_{350179657016286}$
$61312896094022025483863125042 \; = \; P_{202176319243084}$

$260193239740869591046300939711 \; = \; T_{721378180624933}$
$260193239740869591046300939712 \; = \; P_{416487886771328}$

$11894385787986237116124925856566 \; = \; T_{4877373430031011}$
$11894385787986237116124925856567 \; = \; P_{2815952862766733}$

$50476147271823947967491237764746 \; = \; T_{10047501905630468}$
$50476147271823947967491237764747 \; = \; P_{5800927929899029}$

$2307449529973235978502751753048875 \; = \; T_{67933048363417874}$
$2307449529973235978502751753048876 \; = \; P_{39221163759491176}$

$9792112377494105036102253825421125 \; = \; T_{139943648498201625}$
$9792112377494105036102253825421126 \; = \; P_{80796503131815076}$