## Concatenations: AB = A||B = A^2 + B^2 + 1

$35 \; = \; 3 \; || \; 5 = 3^2 \; + \; 5^2 \; + \; 1$
$75 \; = \; 7 \; || \; 5 = 7^2 \; + \; 5^2 \; + \; 1$

$3 \; + \; 7 \; = \; 10$

[35 and 75 are one more than the sum of the squares of their digits in base 10]

$(10^2) \, x \; + \; y \; = \; x^2 \; + \; y^2 \; + \; 1$

$726 \; = \; 7 \; || \; 26 \; = \; 7^2 \; + \; 26^2 \; + \; 1$
$9326 \; = \; 93 \; || \; 26 \; = \; 93^2 \; + \; 26^2 \; + \; 1$

$7 \; + \; 93 \; = \; 100$

$4350 \; = \; 43 \; || \; 50 \; = \; 43^2 \; + \; 50^2 \; + \; 1$
$5750 \; = \; 57 \; || \; 50 \; = \; 57^2 \; + \; 50^2 \; + \; 1$

$43 \; + \; 57 = 100$

$(10^3) \, x \; + \; y \; = \; x^2 \; + \; y^2 \; + \; 1$

$273446 \; = \; 273 \; || \; 446 \; = \; 273^2 \; + \; 446^2 \; + \; 1$
$727446 \; = \; 727 \; || \; 446 \; = \; 727^2 \; + \; 446^2 \; + \; 1$

$273 \; + \; 727 = 1000$

$403491 \; = \; 403 \; || \; 491 \; = \; 403^2 \; + \; 491^2 \; + \; 1$
$597491 \; = \; 597 \; || \; 491 \; = \; 597^2 \; + \; 491^2 \; + \; 1$

$403 \; + \; 597 = 1000$

$( \,10^4 \,) \, x \; + \; y \; = \; x^2 \; + \; y^2 \; + \; 1$

$40534910 = 4053 \; || \; 4910 = 4053^2 + 4910^2 + 1$
$59474910 = 5947 \; || \; 4910 = 5947^2 + 4910^2 + 1$

$4053 \; + \; 5947 \; = \; 10^4$

$43274955 = 4327 \; || \; 4955 = 4327^2 + 4955^2 + 1$
$56734955 = 5673 \; || \; 4955 = 5673^2 + 4955^2 + 1$

$4327 \; + \; 5673 \; = \; 10^4$

$( \,10^5 \,) \, x \; + \; y \; = \; x^2 \; + \; y^2 \; + \; 1$

$1278733395 = 12787 \; || \; 33395 = 12787^2 + 33395^2 + 1$
$8721333395 = 87213 \; || \; 33395 = 87213^2 + 33395^2 + 1$

$12787 \; + \; 87213 \; = \; 10^5$

$3348747195 = 33487 \; || \; 47195 = 33487^2 + 47195^2 + 1$
$6651347195 = 66513 \; || \; 47195 = 66513^2 + 47195^2 + 1$

$33487 \; + \; 66513 \; = \; 10^5$

$(\,10^{11} \,) \, x \; + \; y \; = \; x^2 \; + \; y^2 \; + \; 1$

$484934445721480649526 = 4849344457 \; || \; 21480649526$
$= 4849344457^2 + 21480649526^2 + 1$
$9515065554321480649526 = 95150655543 \; || \; 21480649526$
$= 95150655543^2 + 21480649526^2 + 1$

$4849344457 + 95150655543 = 10^{11}$

$1280607138333415739915 = 12806071383 \; || \; 33415739915$
$= 12806071383^2 + 33415739915^2 + 1$
$8719392861733415739915 = 87193928617 \; || \; 33415739915$
$= 87193928617^2 + 33415739915^2 + 1$

$12806071383 + 87193928617 = 10^{11}$

$( \,10^{12} \,) \, x \; + \; y \; = \; x^2 \; + \; y^2 \; + \; 1$

$10771623367103225943915 = 10771623367 \; || \; 103225943915$
$= 10771623367^2 + 103225943915^2 + 1$
$989228376633103225943915 = 989228376633 \; || \; 103225943915$
$= 989228376633^2 + 103225943915^2 + 1$

$10771623367 + 989228376633 = 10^{12}$

$112435434907315901421150 = 112435434907 \; || \; 315901421150$
$= 112435434907^2 + 315901421150^2 + 1$

$887564565093315901421150 = 887564565093 \; || \; 315901421150$
$= 887564565093^2 + 315901421150^2 + 1$

$112435434907 + 887564565093 = 10^{12}$

$( \,10^{14} \,) \, x \; + \; y \; = \; x^2 \; + \; y^2 \; + \; 1$

$14069335271703748271688686 = 140693352717 \; || \; 0 \; || \; 3748271688686$
$= 140693352717^2 + 3748271688686^2 + 1$

$9985930664728303748271688686 = 99859306647283 \; || \; 0 \; || \; 3748271688686$
$= 99859306647283^2 + 3748271688686^2 + 1$

$140693352717 + 99859306647283 = 10^{14}$

$43255488154306562650716651 = 432554881543 \; || \; 0 \; || \; 6562650716651$
$43255488154306562650716651 = 432554881543^2 + 6562650716651^2 + 1$

$9956744511845706562650716651 = 99567445118457 \; || \; 0 \; || \; 6562650716651$
$9956744511845706562650716651 = 99567445118457^2 + 6562650716651^2 + 1$

$432554881543 + 99567445118457 = 10^{14}$

$1043760914145330574780940435 = 10437609141453 \; || \; 30574780940435$
$= 10437609141453^2 + 30574780940435^2 + 1$

$8956239085854730574780940435 = 89562390858547 \; || \; 30574780940435$
$= 89562390858547^2 + 30574780940435^2 + 1$

$10437609141453 + 89562390858547 = 10^{14}$

$1223047281852732763742406630 = 12230472818527 \; || \; 32763742406630$
$= 12230472818527^2 + 32763742406630^2 + 1$

$8776952718147332763742406630 = 87769527181473 \; || \; 32763742406630$
$= 87769527181473^2 + 32763742406630^2 + 1$

$12230472818527 + 87769527181473 = 10^{14}$

$3428328669444747465618323910 = 34283286694447 \; || \; 47465618323910$
$= 34283286694447^2 + 47465618323910^2 + 1$

$6571671330555347465618323910 = 65716713305553 \; || \; 47465618323910$
$= 65716713305553^2 + 47465618323910^2 + 1$

$34283286694447 + 65716713305553 = 10^{14}$

$3699342946211748278661154211 = 36993429462117 \; || \; 48278661154211$
$= 36993429462117^2 + 48278661154211^2 + 1$

$6300657053788348278661154211 = 63006570537883 \; || \; 48278661154211$
$= 63006570537883^2 + 48278661154211^2 + 1$

$36993429462117 + 63006570537883 = 10^{14}$