Numbers 660,000,000,000s

660,790,152,100

[Math.]  35,010,152,100 = (35010+152,100)2 = 187,1102 and

660,790,152,100 = (660,790+152,100)2 = 812,8902 and 187110+812890 = 1,000,000.

661,700,902,500

[Math.]  Sum of the first 1275 cube numbers:

13+…+12753 = 661,700,902,500 = (1+…+1275)2 = 813,4502.

663,574,218,751

[Math.]  Its odd powers end with 663,574,218,751. Its even powers end with 000,000,000,001.

663,778,455,076

[Math.]  Sum of the first 1276 cube numbers:

13+…+12763 = 663,778,455,076 = (1+…+1276)2 = 814,7262.

664,891,837,281

[Math.]  (1+2+…+42)4 = [(15+25+…+425)+(17+27+…+427)]/2.

665,860,896,009

[Math.]  Sum of the first 1277 cube numbers:

13+…+12773 = 665,860,896,009 = (1+…+1277)2 = 816,0032.

665,947,081,800

[Math.]  Two consecutive sums of consecutive square numbers:

665,947,081,800 = 614252+…+616002 = 616012+…+617752.

666,666,333,334

[Math.]

642 = 4096 = 4095+1

66342 = 44009956 = 44009955+1

666,3342 = 444,000,999,556 = 444,000,999,555+1

666633342 = 4444000099995556 = 4444000099995555+1

66666333342 = 44444000009999955556 =  44444000009999955555+1

6666663333342 = 444444000000999999555556 = 444444000000999999555555+1.

666,666,666,665

[Math.]  Repeatedly inserting numbers 42 between the digits 4 and 2 in the square number 4225 = 652 yields other squares 442225 = 6652, 44422225 = 66652, 444222225 = 666652

[Math.]  Square of numbers of similar pattern xxy, where y = x–1.

*  1102 = 012,100 & 012+100 = 112 = 110+2

*  3322 = 110,224 & 110+224 = 334 = 332+2.

Similarly, 3…322 = 1…102…24 & 1…10+2…24 = 3…34 = 3…32+2

*  4432 = 196,249 & 196+249 = 445 = 443+2.

*  6652 = 442,225 & 442+225 = 667 = 665+2.

Similarly, 6…652 = 4…422…25 & 4…42+2…25 = 6…67 = 6…65+2

66,666,666,6652 = 4444444444222222222225 & 44444444442+22222222225 = 66,666,666,667 = 66,666,666,665+2.

*  7762 = 602,176 & 602+176 = 778 = 776+2.

666,666,666,666

[Math.]

666,666,666,666×666,666,666,664 = 444,444,444,442,222,222,222,224

666,666,666,666×666,666,666,667 = 444,444,444,444,222,222,222,222

666,666,666,666×666,666,666,668 = 444,444,444,444,888,888,888,888

666,666,666,666×666,666,666,669 = 444,444,444,445,555,555,555,554.

666,666,666,666.3333333333332…

[Math.]  The square root of the number formed by n digits 4 followed by n digits 0 starts with n digits 6, decimal point, and n digits 3:

401/2 = 6.32…

44001/2 = 66.332…

4440001/2 = 666.3332…

444400001/2 = 6666.33332…

44444000001/2 = 66666.333332…

4444440000001/2 = 666666.3333332…

444444400000001/2 = 6666666.33333332…

44444444000000001/2 = 66666666.333333332…

4444444440000000001/2 = 666666666.3333333332…

44444444440000000001/2 = 6666666666.33333333332…

4444444444400000000001/2 = 66666666666.333333333332…

444444444444000000000001/2 = 666666666666.3333333333332…

44444444444440000000000001/2 = 6666666666666.33333333333332…

4444444444444400000000000001/2 = 66666666666666.333333333333332…

666,666,666,667

[Math.]  333333333333,666666666667,000000000000 =

333,333,333,3333+666,666,666,6673+0000000000003

and 333333333333,666666666667,000000000001 =

333,333,333,3333+666,666,666,6673+0000000000013. (Generalizable)

1,333,333,333,334,666,666,666,667 =

= 1,333,333,333,3342–666,666,666,6672 =

= (133333333334+666666666667)×666666666667. (Generalizable)

333,333,333,334,000,000,000,000,666,666,666,667 =

= 333,333,333,3343+0000000000003+666,666,666,6673 and

333,333,333,334,000,001,000,000,666,666,666,667 =

= 333,333,333,3343+0000010000003+666,666,666,6673. (Generalizable)

[Math.]  Repeatedly inserting numbers 48 in the middle of the square number 49 yields other squares 4489 = 672, 444889 = 6672, 44448889 = 66672, 4444488889 = 666672

666,666,666,668

[Math.]  3,333,333,333,466,666,666,668 = 666,666,666,6682–333,333,333,3342  =  (333333333334+666666666668)×333333333334. (Generalizable)

667,948,232,961

[Math.]  Sum of the first 1278 cube numbers:

13+…+12783 = 667,948,232,961 = (1+…+1278)2 = 817,2812.

669,420,148,761

[Math.]  33,058,148,761 = (33,058+148,761)2 = 181,8192 and

669,420,148,761 = (669420+148761)2 = 818,1812 and 181819+818181 = 1,000,000.