Numbers 660,000s

A page of the Numeropedia - the Special Encyclopedia of Numbers

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660,659 & 660,661

[Math.]  600,659 and 660,661 are twin prime numbers

 

660,790

[Math.]  35,010,152,100 = (35010+152,100)2 = 187,1102 and 660,790,152,100 = (660790+152100)2 = 812,8902 and 187110+812890 = 1,000,000.

 

663,552

[Math.]  663,552 = 12×23×34×45.

 

664,579

[Math.]  A prime number.

 

[Math.]  The number of prime numbers less than 10,000,000.

 

666,334

[Math.]  

642 = 4096 = 4095+1

66342 = 44009956 = 44009955+1

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

 

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, 4444222225 = 666652, 444442222225 = 6666652

 

[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.

666,6652 = 444,442,222,225 & 444,442+222,225 = 666,667 = 666,665+2.

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

 

666,666

[Math.]  The area of the primitive Pythagorean triangle of sides (693, 1924, 2045).

 

[Math.] 

 121×(1+2+1) = 484 = 222

12,321×(1+2+3+2+1) = 110,889 = 3332

12,34,321×(1+2+3+4+3+2+1) = 19,749,136 = 4,4442

123,454,321×(1+2+3+4+5+4+3+2+1) = 3,086,358,025 = 55,5552

12,345,654,321×(1+2+3+4+5+6+5+4+3+2+1) = 444,443,555,556 = 666,6662

1,234,567,654,321×(1+2+3+4+5+6+7+6+5+4+3+2+1) =

= 60,493,815,061,729 = 7,777,7772

123,456,787,654,321×(1+2+3+4+5+6+7+8+7+6+5+4+3+2+1) =

= 7,901,234,409,876,544 = 88,888,8882

12,345,678,987,654,321×(1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1) =

= 999,999,998,000,000,001 = 999,999,9992.

 

[Trivia]  An apocalyptic number is defined as a number of the form 2n that contains the digits 666. A double apocalyptic number is defined as a number of the form 2n that contains the digits 666,666. 

 

666,666.3333332…

[Math.] 

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…

...

 

666,667

[Math.]  A prime number.

333333,666667,000000 = 333,3333+666,6673+0000003 and

333333,666667,000001 = 333,3333+666,6673+0000013.

 

[Math.]  1,333,334,666,667 = 1,333,3342–666,6672 = (1333334+666667)×666667,

where 133334 = 666667×2.

 

[Math.]  333,334,000,000,666,667 = 3333343+0000003+6666673 and

333,334,001,000,666,667 = 3333343+0010003+6666673. (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

 

666,668

[Math.]   333,334,666,668 = 666,6682–333,3342 =

= (333334+666668)×333,334, where 666668 = 333334×2.

 

667,964

[Math.]  A pair of amicable numbers (667964, 783556).

 

669,420

[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.

 

[Math.]  Sum (in degrees) of all internal angles of a 3721-side polygon.

 

669,688

[Math.]  A pair of amicable numbers (600392, 669688).

 

669,900

[Math.]  Two consecutive sums of consecutive numbers:

7569+…+7656 = 7657+…+7743 = 669,900

 

666661, 666663, 666664, 666666, 666667, 666668666669

666670, 666671, 666672, 666674 & 666675

[Math.] 

66×64 = 4224

66×67 = 4422

66×68 = 4488

66×69 = 4554

66×74 = 4884

...

666666×666661 = 444440,222226

666666×666669 = 444445,555554

666666×666663 = 444441,555558

666666×666670 = 444446,222220

666666×666664 = 444442,222224

666666×666671 = 444446,888886

666666×666666 = 444443,555556

666666×666672 = 444447,555552

666666×666667 = 444444,222222

666666×666674 = 444448,888884

666666×666668 = 444444,888888

666666×666675 = 444449,555550

… (generalizable pattern) …