8,101,265,822,784

[Math.]  8,101,265,822,784 = 8×1,012,658,227,848.

8,756,963,649,152

[Math.]  The 49th Keith number: 8,756,963,649,152.

2,199,999,999,978 & 4,399,999,999,956 & 8,799,999,999,912

[Math.]

(*)   8712 = 2178×4 & 8712×2178 = 18,974,736 = 43562

87128712 = 21782178×4 & 87128712×21782178 = 435643562

(**)   87912 = 21978×4 & 21978×87912 = 1,932,129,936 = 439562

8791287912 = 21978…21978×4 & 21978…21978×87912…87912 = 43956…439562

...

(***)   8,799,999,999,912 = 2,199,999,999,978×4 &

2,199,999,999,978×8,799,999,999,912 = 4,399,999,999,9562

8799...99128799...9912 = 2199...99782199...9978×4 &

2199...99782199...9978×8799...99128799...9912 = 4399...99564399...99562

...

(****)   879…912 = 219…978×4 & 219…978×879…912 = 439…9562

879…912879…912 = 219…978…219…978×4 &

219…978219…978×879...912…879...912 = 439…956439…9562.

8,822,750,406,821 & 6,238,626,641,379 & 6,238,626,641,380

[Math.]  The 17th square number that is the sum of squares of 2 consecutive numbers: 6,238,626,641,3792+6,238,626,641,3802= 8,822,750,406,8212. Each root number in the series is 6 times the previous root number minus the earlier root number: 1, 52, 292, 1692, 9852

Or in other words:

(6,238,626,641,379, 6,238,626,641,380, 8,822,750,406,821) is the 17th Pythagorean triple (lengths of 3 sides of a right triangle), where 2 sides are consecutive integers.

8,888,888,888,888

[Math.]

8×8+13 = 77

88×8+13 = 717

888×8+13 = 7117

8888×8+13 = 71117

88,888×8+13 = 711,117

888,888×8+13 = 7,111,117

8,888,888×8+13 = 71,111,117

88,888,888×8+13 = 711,111,117

888,888,888×8+13 = 7,111,111,117

8,888,888,888×8+13 = 71,111,111,117

88,888,888,888×8+13 = 711,111,111,117

888,888,888,888×8+13 = 7,111,111,111,117

8,888,888,888,888×8+13 = 77,111,111,111,117

88,888,888,888,888×8+13 = 777,111,111,111,117

...

[Math.]

9×9+(9–2) = 88

98×9+(9–3) = 888

987×9+(9–4) = 8,888

9,876×9+(9–5) = 88,888

98,765×9+(9–6) = 888,888

987,654×9+(9–7) = 8,888,888

9,876,543×9+(9–8) = 88,888,888

98,765,432×9+(9–9) = 888,888,888

987,654,321×9+(9–10) = 8,888,888,888

9,876,543,210×9+(9–11) = 88,888,888,888

98,765,432,099×9+(9–12) = 888,888,888,888  = 98,765,432,100×9–12.

987,654,320,988×9+(9–13) = 8,888,888,888,888

9,876,543,209,877×9+(9–14) = 88,888,888,888,888

98,765,432,098,766×9+(9–15) = 888,888,888,888,888

987,654,320,987,655×9+(9–16) = 8,888,888,888,888,888

9,876,543,209,876,544×9+(9–17) = 88,888,888,888,888,888

...

8,916,100,448,256