Do You Know

31 December, 2023:

• The only way to express three fractions containing just the nine digits to have a sum of 1 is 9/12 + 7/68 + 5/34 = 1.

12 July, 2023:

• 567 and 854 are the only two numbers m, such that m and m2 contain each of the digits 1 to 9 only once, as shown below:
5672= 321489, and, 8542= 729316

12 September, 2022:

• √764 = 27.64054992...
√76394 = 276.3946453895...
√7639321 = 2763.9321626986...
√763932023 = 27639.3202340433...

14 March, 2022:

• Except 1 and 9, there is no perfect square with all of its digits odd.

11 November, 2021:

• The number of letters in the thirteen cards of a suit i.e. ACE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN, JACK, QUEEN and KING is 52 which is the number of cards in the standard deck of playing cards.

12 April, 2021:

• 216 = 65536 is the only known power of 2 in which no digit is a power of 2.

17 June, 2020:

• The product of four consecutive positive integers can not be equal to product of two consecutive positive integers.

14 December, 2019:

• The number 665067264 is the largest even number such that the number formed from first n digits is divisible by nthprime.

15 September, 2019:

• The digital root of product of twin primes other than (3,5) is always 8.

28 November, 2018:

• There can not be three consecutive triangular numbers which can form a Pythagorean triple.

17 December, 2017:

• 270 is the smallest positive integer such that it has divisors ending with every decimal digit i.e. divisors ending in 0,1,2,3,4,5,6,7,8 and 9.

30 October, 2016:

• 1/37 = 0.027027027...
1/27 = 0.037037037...

(Comments by Al Zimmermann dt 04-05-2017 )

The decimal expansions for 1/27 and 1/37 are as shown above because 27 × 37 = 999.
Another example would be:
1/271 = .00369 00369 00369 …
1/369 = .00271 00271 00271 …
because 271 × 369 = 99999.

4 November, 2015:

• There can not be four triangular numbers in arithmetical progression.

7 July, 2015:

• 541993 = 159211275242599 and 15921 + 12752 + 42599 = 71272
712723 = 362040234715648 and 36204 + 02347 + 15648 = 54199

6 October, 2014:

• 48 = 82 - 42
484848 = 8482 - 4842
4848484848 = 848482 - 484842
48484848484848 = 84848482 - 48484842and so on.

25 December, 2013:

• There are 153 days in any five consecutive months not containing February.

15 July, 2013:

• It is impossible to construct a triangle with sides as distinct Fibonacci numbers

13 November, 2012:

• The smallest perfect number 6 = 1 x 2 x 3
The smallest multiply perfect number 120 = 4 x 5 x 6
The SUM of smallest pair of AMICABLE numbers 220 + 284 = 504 = 7 x 8 x 9

6 February, 2011:

• The smallest integer n such that n/2 is perfect square, n/3 is perfect cube and n/5 is perfect fifth power is 30233088000000.

26 January, 2010:

• The number of letters in ODD, EVEN, PRIME and COMPOSITE are odd, even, prime and composite respectively.

10 May, 2009:

• Every even number greater than 46 can be expressed as sum of two abundant numbers.

18 January, 2009:

• 639172 is the largest integer with distinct digits whose square consists of digits not included in itself i.e. 639172² = 408,540,845,584

30 December, 2007:

• The Golden Ratio (Phi=1.61803...) can be expressed using four 4's as (√ 4 + √ (4!-4) )/4.

14 July, 2007:

• 1, 6 and 120 are the only numbers which are both triangular and factorial.

26 January, 2007:

• 8589934592 x 116415321826934814453125 (No zero in both numbers)
= 1000000000000000000000000000000000
This is the largest known example of this kind.

25 October, 2006:

• The fraction 1/998999 contains Fibonacci numbers i.e.

1/998999=0.000001001002003005008013021034055089..

(Comments by Al Zimmermann dt 01-05-2017 )

This fraction contains Fibonacci numbers up to 3 digits in length.
This can be generalized as follows::
The fraction 1/(10^2n-10^n-1) contains Fibonacci numbers up to n digits in length.
If n = 3, the fraction is 1 / (1000000 – 1000 – 1) = 1/998999.
For n = 4, the fraction is 1/99989999 = .0000000100010002000300050008001300210034005500890144….
The n leading zeroes in the decimal expansion can be considered the 0th term in the Fibonacci series or they can be removed by changing the fractions numerator to 10^n:
10000/99989999 = .000100010002000300050008001300210034005500890144….

15 January, 2006:

• The product of 4 consecutive natural numbers can not be a cube.

7 November, 2005:

• A multidigit number with all its digits odd can never be a perfect square.

1 May, 2005:

• 27, 58 and 85 are three consecutive Smith numbers and 27 + 58 = 85 .

11 March, 2005:

• The number 3435 is the number such that 3435 = 3^3 +4^4 + 3^3 + 5^5 .

1 January, 2005:

• The number 3608528850368400786036725 is the largest number such that the number formed from first n digits is divisible by n.

15 September, 2004:

• The number FORTY is the only number in English language, all letters of which are in alphabetical order.

4 May, 2004:

• The nth fibonacci number Fn is divisible by 9, if and only if, n and Fn both are even.

2 March, 2004:

• The product of all divisors of a number n is sqrt(n^d), where d is no. of divisors of n.

1 February, 2004:

• The only number which is equal to the sum of subfactorial of its digits is 148349. (i.e. 148349 = !1+!4+!8+!3+!4+!9).

1 January, 2004:

• The only known pair of Twin Pseudoprimes(base-2) is 4369 and 4371.

8 November, 2003:

• Every Carmichael number is squarefree.

3 August, 2003:

• No Fibonacci number is equal to the product of two smaller Fibonacci numbers.

13 June, 2003:

• 1 and 6 are the only triangular Numbers whose squares are also triangular numbers.

21 May, 2003:

• Every number greater than 11 is the sum of two composite numbers.

29 March, 2003:

• 8549176320 is a curious number(containing all ten digits) whose digits are in alphabetical order.

1 March, 2003:

• In decimal system 6661661161 is the largest known square with two distinct non-zero digits.

1 February, 2003:

• 10662526601 is the only known palindromic cube such that its cube root (i.e. 2201) is non-palindromic.

1 January, 2003:

• 8114118 is the smallest multidigit palindrome such that 8114118th prime i.e. 143787341 is also palindrome.

1 December, 2002:

• Two consecutive numbers i.e. n and n+1 are always relatively prime.

11 November, 2002:

• The largest Triangular Number which is the product of three consecutive numbers is 258474216.

5 October, 2002:

• All Semiprimes are deficient numbers.

21 September, 2002:

• The smallest Fibonacci number containing all digits from 0 to 9 is F74 i.e. 1304969544928657.

6 September, 2002:

• Triangular numbers can never end in 2,4,7 or 9.

14 April, 2002:

• The digit in the tens place of a power of 7 is always 0 or 4.

10 March, 2002:

• All Composite Mersenne numbers are Strong Pseudoprimes (base-2).

10 February, 2002:

• There can be maximum five consecutive deficient Numbers and smallest such set is 7, 8, 9, 10 and 11.

20 January, 2002:

• There can not be four perfect squares in arithmetical progression.

6 January, 2002:

• The smallest triplet of consecutive Abundant Numbers is 171078830, 171078831, 171078832.

31 December, 2001:

• The largest known Fibonacci Number consisting of only odd digits is 17711.

23 December, 2001:

• 41096 * 83 = 3410968

17 December, 2001:

• The smallest Fibonacci Number that is also a Smith Number is 1346269.

9 December, 2001:

• The numbers in the series 1, 11, 111, 1111, 11111, ... are all Triangular Numbers in base-9.

30 November, 2001:

• The smallest titanic factorial is 450! . This has 1001 digits and is also known as Arabian Nights Factorial.

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