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

 
 
Heptagonal numbers are of the form    n \,(5 \, n - 3)/2

 

We can write

10^{n} \; x \; + \; (x + 1) \; = \; y \,(5 \, y - 3)/2

Or

(10^{n} + 1) \, x \; + \; 1 \; = \; y \,(5 \, y - 3)/2

the integer length of   x   is   n

 
Here are the first few solutions
 

n = 1

3 \; || \; 4 \; = \; 34 \; = \; H_4

n = 2

39 \; || \; 40 \; = \; 3940 \; = \; H_{40}

n = 3

238 \; || \; 239 \; = \; 238239 \; = \; H_{309}
399 \; || \; 400 \; = \; 399400 \; = \; H_{400}

n = 4

1521 \; || \; 1522 \; = \; 15211522 \; = \; H_{2467}
3999 \; || \; 4000 \; = \; 39994000 \; = \; H_{4000}

n = 5

39999 \; || \; 40000 \; = \; 3999940000 \; = \; H_{40000}
51655 \; || \; 51656 \; = \; 5165551656 \; = \; H_{45456}

n = 6

183991 \; || \; 183992 \; = \; 183991183992 \; = \; H_{271287}
399999 \; || \; 400000 \; = \; 399999400000 \; = \; H_{400000}

n = 7

1190082 \; || \; 1190083 \; = \; 11900821190083 \; = \; H_{2181818}
3999999 \; || \; 4000000 \; = \; 39999994000000 \; = \; H_{4000000}

n = 8

39999999 \; || \; 40000000 \; = \; 3999999940000000 \; = \; H_{40000000}
55363324 \; || \; 55363325 \; = \; 5536332455363325 \; = \; H_{47058825}

n = 9

173130195 \; || \; 173130196 \; = \; 173130195173130196 \; = \; H_{263157896}
178399921 \; || \; 178399922 \; = \; 178399921178399922 \; = \; H_{267132867}
330578514 \; || \; 330578515 \; = \; 330578514330578515 \; = \; H_{363636365}
337845235 \; || \; 337845236 \; = \; 337845235337845236 \; = \; H_{367611336}
392089558 \; || \; 392089559 \; = \; 392089558392089559 \; = \; H_{396025029}
399999999 \; || \; 400000000 \; = \; 399999999400000000 \; = \; H_{400000000}
982177609 \; || \; 982177610 \; = \; 982177609982177610 \; = \; H_{626794260}
994674555 \; || \; 994674556 \; = \; 994674555994674556 \; = \; H_{630769231}

n = 10

1921380258 \; || \; 1921380259 \; = \; 19213802581921380259 \; = \; H_{2772277229}
2861552983 \; || \; 2861552984 \; = \; 28615529832861552984 \; = \; H_{3383225079}
3999999999 \; || \; 4000000000 \; = \; 39999999994000000000 \; = \; H_{4000000000}
9472552161 \; || \; 9472552162 \; = \; 94725521619472552162 \; = \; H_{6155502307}

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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