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}

 

 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s