Do You Know

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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