10,000,000,000,037 [Math.] The smallest 14-digit prime number. 10,999,999,999,989 & 32,999,999,999,967 & 98,999,999,999,901 [Math.] (*) 332×9 = 1089×9 = 9,801 = 992 & 1089×9801 = 10,673,289 = 32672 1089…1089×9 = 9801…9801 & 1089…1089×9801…9801 = 3267…32672 (**) 10989×9 = 98901 & 10989×98901 = 1,086,823,089 = 329672 10989…10989×9 = 98901…98901 & 10989…10989×98901...98901 = 32967…329672 ... (***) 10,999,999,999,989×9 = 98,999,999,999,901 & 10,999,999,999,989×98,999,999,999,901 = 32,999,999,999,9672 1099999999989…109999999989×9 = 9899999999901…9899999999901 & 1099999999989…1099999999989×9899999999901...9899999999901 = 32999999967…32999999999672 ... (****) 109…989×9 = 989…901 & 109...989×989…901 = 329…9672 109…989...109…989×9 = 989…901...989…901 & 109…989…109…989×989…901...989…901 = 329…967…329…9672.
11,223,344,332,211 [Math.] 11×11 = 121; 112211×11 = 1,234,321 1122332211×11 = 12,345,654,321 11223344332211×11 = 123,456,787,654,321 112,233,445,544,332,211×11 = 1,234,567,900,987,654,321 …
12,345,678,901,234 & 98,765,432,109,876 [Math.] 1×8+1 = 9 12×8+2 = 98 123×8+3 = 987 1234×8+4 = 9876 12,345×8+5 = 98,765 123,456×8+6 = 987,654 1,234,567×8+7 = 9,876,543 12,345,678×8+8 = 98,765,432 123,456,789×8+9 = 987,654,321 1,234,567,890×8+90 = 9,876,543,210 (Start of new cycle) 12,345,678,901×8+901 = 98,765,432,109 123,456,789,012×8+9002 = 987,654,321,098 1,234,567,890,123×8+90,003 = 9,876,543,210,987 12,345,678,901,234×8+900,004 = 98,765,432,109,876 123,456,789,012,345×8+9,000,005 = 987,654,321,098,765 1,234,567,890,123,456×8+90,000,006 = 9,876,543,210,987,654 12,345,678,901,234,567×8+900,000,007 = 98,765,432,109,876,543 123,456,789,012,345,678×8+9,000,000,008 = 987,654,321,098,765,432 1,234,567,890,123,456,789×8+90,000,000,009 = 9,876,543,210,987,654,321 12,345,678,901,234,567,890×8+ 900,000,000,090 = 98,765,432,109,876,543,210 … (end of cycle)
13,689,992,600,001 & 36,999,999,999,999 & 33,669,999,999,999 [Math.] (See below) 14,285,714,285,714 [Math.] 1×7 + 3 = 10 14×7 + 2 = 100 142×7 + 6 = 1000 1428×7 + 4 = 10000 14285×7 + 5 = 100000 142857×7 + 1 = 1000000
1428571×7 + 3 = 10000000 14285714×7 + 2 = 100000000 142857142×7 + 6 = 1000000000 1428571428×7 + 4 = 10000000000 14285714285×7 + 5 = 100000000000 142857142857×7 + 1 = 1000000000000
1428571428571×7 + 3 = 10000000000000 14285714285714×7 + 2 = 100000000000000 142857142857142×7 + 6 = 1000000000000000 1428571428571428×7 + 4 = 10000000000000000 14285714285714285×7 + 5 = 100000000000000000 142857142857142857×7 + 1 = 1000000000000000000 … 15,999,999,999,984 & 27,999,999,999,972 & 48,999,999,999,951 [Math.] 7,683,984 = 1584×4851 = 27722 782,432,784 = 15984×48951 = 279722 78,384,320,784 = 159984×489951 = 2799722 7,839,843,200,784 = 1599984×4899951 = 27999722 783,998,432,000,784 = 15999984×48999951 = 279999722 78,399,984,320,000,784 = 159999984×489999951 = 2799999722 7,839,999,843,200,000,784 = 1599999984×4899999951 = 27999999722 783,999,998,432,000,000,784 = 15999999984×48999999951 = 279999999722 78,399,999,984,320,000,000,784 = 159999999984×489999999951 = 2799999999722 … 7839…9984320000784 = 159…999984×489…999951 = 279…9999722. 19,359,612,801,936 & 21,999,999,999,978 & 43,999,999,999,956 & 87,999,999,999,912
[Math.] (*) 8712 = 2178×4 & 8712×2178 = 18,974,736 = 43562 8712…8712 = 2178…2178×4 & 8712…8712×2178…2178 = 4356…43562
(**) 87912 = 21978×4 & 21978×87912 = 1,932,129,936 = 439562 87912…87912 = 21978…21978×4 & 21978…21978×87912…87912 = 43956…439562 ... (***) 8799912 = 2199978×4 & 2199978×8799912 = 19,359,612,801,936 = 43999562 8799912…8799912 = 2199978…2199978×4 & 2199978…2199978×8799912…8799912 = 4399956…43999562 ... (***) 87,999,999,999,912 = 21,999,999,999,978×4 & 21,999,999,999,978×87,999,999,999,912 = 43,999,999,999,9562 8799...9912…8799...9912 = 2199...9978…2199...9978×4 & 2199...9978…2199...9978×8799...9912…8799...9912 = 4399...9956…4399...99562 ... (****) 879…912 = 219…978×4 & 219…978×879…912 = 439…9562 879…912…879…912 = 219…978…219…978×4 & 219…978…219…978×879...912…879...912 = 439…956…439…9562. 30.861…×1012 [Astronomy] The approximate length (km, kilometers) of a parsec (parallax second), the distance from the Sun that would be a second of arc as observed from the Earth. 1 parsec = 3.262 light years.
31.536×1012 [EECS] A 1-MIPS year is equivalent to the computing power of a MIPS computer running for one year = 31.536×1012 arithmetic operations. MIPS = million instructions per second, for measuring speeds of microprocessors. 33,333,333,333,333 & 33,333,336,666,667 [Math.] 3×37 = 111 33×3367 = 111,111 333×333667 = 111,111,111 3333×33336667 = 111,111,111,111 33333×3333366667 = 111,111,111,111,111 333333×333333666667 = 111,111,111,111,111,111 3333333×33333336666667 = 111,111,111,111,111,111,111 33333333×3333333366666667 = 111,111,111,111,111,111,111,111 333333333×333333333666666667 = 111,111,111,111,111,111,111,111,111 3333333333×33333333336666666667 = 111,111,111,111,111,111,111,111,111,111 33333333333×3333333333366666666667 = = 111,111,111,111,111,111,111,111,111,111,111 333333333333×333333333333666666666667 = = 111,111,111,111,111,111,111,111,111,111,111,111
3333333333333×33333333333336666666666667 = = 111,111,111,111,111,111,111,111,111,111,111,111,111 33,333,333,333,333×3333333333333366666666666667 = = 111,111,111,111,111,111,111,111,111,111,111,111,111,111 ... 29,999,999,999,996 & 33,333,336,666,667 & 99,987,666,543,332
[Math.] 296×333,667 = 98,765,432 2,996×33,336,667 = 99,876,654,332 29,996×3,333,366,667 = 99,987,666,543,332 299,996×333,333,666,667 = 99,998,766,665,433,332 2,999,996×33,333,336,666,667 = 99,999,876,666,654,333,332 29,999,996×3,333,333,366,666,667 = 99,999,987,666,666,543,333,332 299,999,996×333,333,333,666,666,667 = 99,999,998,766,666,665,433,333,332 2,999,999,996×33,333,333,336,666,666,667 = 99,999,999,876,666,666,654,333,333,332 29,999,999,996×3,333,333,333,366,666,666,667 = = 99,999,999,998,766,666,666,663,333,333,332 299,999,999,996×333,333,333,333,366,666,666,667 = = 99,999,999,998,766,666,666,665,433,333,333,332 ...
33,333,336,666,667
[Math.] 372 = 1369 = 1368+1; 33672 = 11336689 = 11336688+1 333,6672 = 111,333,666,889 = 111,333,666,888+1 333366672 = 1111333366668889 = 1111333366668888+1 33333666672 = 11111333336666688889 = 11111333336666688888+ 3333336666672 = 111111333333666666888889 = 111111333333666666888888+1 333333366666672 = = 1111111333333366666668888889 = 1111111333333366666668888888+1 … 13,689,992,600,001 & 36,999,999,999,999 & 33,669,999,999,999 [Math.] 3692 = 136,161 36992 = 13682601 36,9992 = 1368926001 369,9992 = 136899260001 3,699,9992 = 13689992600001 36,999,9992 = 1368999926000001 369,999,9992 = 136899999260000001 3,699,999,9992 = 13689999992600000001 36,999,999,9992 = 1368999999926000000001 369,999,999,9992 = 136899999999260000000001 3366992 = 113366,216601 336699992 = 11336688,32660001 33669999992 = 1133668899,3266000001 3366999999992 = 113366889999,326600000001 … 333,666,9992 = 111,333,666,221,666,001 333,666,999,9992 = 111,333,666,888,332,666,000,001 333,666,999,999,9992 = 111,333,666,888,999,332,666,000,000,001 … 3333,6666,99992 = 1111,3333,6666,2221,6666,0001 3333,6666,9999,99992 = 1111,3333,6666,8888,3332,6666,0000,0001 … 36,361,380,737,780 & 36,361,380,737,781 & 51,422,757,785,981 [Math.] The 18th square number that is the sum of squares of 2 consecutive numbers: 51,422,757,785,9812+51,422,757,785,9812= 51,422,757,785,9812. 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: (36,361,380,737,780, 36,361,380,737,781, 51,422,757,785,981) is the 18th Pythagorean triple (lengths of 3 sides of a right triangle), where 2 sides are consecutive integers. 43,520,999,798,747 [Math.] The 50th Keith number: 43,520,999,798,747.
43,892,069,261,881 [Math.] The 9th square number that is the sum of squares of 2 consecutive numbers: 6,625,1092 = 43,892,069,261,881 = 4,684,6592+4,684,6602. 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: (4,684,659, 4,684,660, 6,625,109) is the 9th Pythagorean triple (lengths of 3 sides of a right triangle), where 2 sides are consecutive integers.
44,444,400,000,000 & 66,666,666,666,666.33333333333333332... [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… 444444444444444000000000000001/2 = 666666666666666.3333333333333332… 44444444444444440000000000000001/2 = 6666666666666666.33333333333333332… …
51,422,757,785,981 (see above)
60,493,815,061,729 [Math.] 121×(1+2+1) = 484 = 222 12,321×(1+2+3+2+1) = 110,889 = 3332 1,234,321×(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.
61,728,399,382,716 Math.]
The 11th triangle number is 66 The 1111th triangle number is 617,716 The 111,111th triangle number is 6,172,882,716 The 11,111,111th triangle number is 61,728,399,382,716 All numbers are palindromic.
66,666,663,333,334 [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. 666666633333342 = 4444444000000099999995555556 = 4444444000000099999995555555+1. … 74,596,893,730,427 [Math.] A prime number. The 51st Keith number: 74,596,893,730,427.
77,111,111,111,117 & 88,888,888,888,888 [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 ... 88,888,888,888,888 & 98,765,432,098,766 [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 ...
97,295,849,958,669 [Math.] The 52nd Keith number: 97,295,849,958,669. 29,999,999,999,996 & 33,333,336,666,667 & 99,987,666,543,332
[Math.] 296×333,667 = 98,765,432 2,996×33,336,667 = 99,876,654,332 29,996×3,333,366,667 = 99,987,666,543,332 299,996×333,333,666,667 = 99,998,766,665,433,332 2,999,996×33,333,336,666,667 = 99,999,876,666,654,333,332 29,999,996×3,333,333,366,666,667 = 99,999,987,666,666,543,333,332 299,999,996×333,333,333,666,666,667 = 99,999,998,766,666,665,433,333,332 2,999,999,996×33,333,333,336,666,666,667 = 99,999,999,876,666,666,654,333,333,332 29,999,999,996×3,333,333,333,366,666,666,667 = = 99,999,999,998,766,666,666,663,333,333,332 299,999,999,996×333,333,333,333,366,666,666,667 = = 99,999,999,998,766,666,666,665,433,333,333,332 ...
10,999,999,999,989 & 32,999,999,999,967 & 98,999,999,999,901 [Math.] (Above) 99,999,999,999,973 [Math.] The largest 14-digit prime number = 99,999,999,999,973. |