Pankaj Khanna
9424810575
At the same time, I am also thinking that 2022 is afterall just a number. Though it is also an interesting number like most other numbers. I have been saying for years: Love thy numbers!
Let us love this number too.
2022 is a Sphenic number apart from being a Polite number. Now what are the Sphenic numbers!?
A Sphenic number is a product abc where a, b, and c are three distinct prime numbers.
The smallest sphenic number is 30 ( 2 × 3 × 5), the product of the smallest three primes. The first few Sphenic numbers are
30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154,165,......230,231,.....1309,1310,1311,...2013, 2014,2015, 2022,.....2035,2037,....
So the next time you will encounter a Sphenic number in year 2035!
Coming back to number 2022, it has three factors 2, 3 and 337.
2022= 2x3x337
All Sphenic numbers have exactly eight factors. The converse does not hold. For example, 24 is not a Sphenic number, but it has exactly eight factors.
All Sphenic numbers are by definition squarefree, because the prime factors must be distinct and their product will not form a square.
The first case of two consecutive sphenic numbers is 230 and 231.
230 = 2×5×23,
231 = 3×7×11.
A few more examples of two consecutive Sphenic Numbers are:
285 and 286
434 and 435
741 and 742
805 and 806
902 and 903
969 and 970
986 and 987
1001 and 1002
1022 and 1023
1309 and 1310
1310 and 1311
.
.
.
.
2054 and 2055
.
.
.
The list goes on and on.
The first case of three consecutive Sphenic numbers is 1309,1310,1311.
1309 = 7×11×17,
1310 = 2×5×131, and
1311 = 3×19×23.
There is no case of more than three consecutive Sphenic Numbers, because one of the four consecutive positive integers is always divisible by 4 (2×2) and therefore not a squarefree and hence not a Sphenic number.
I remember having informed my students on January 1, 2013 about three consecutive Sphenic numbers 2013, 2014 and 2015. Some of them were delighted. Others, who were in majority, were still in the hangover of 31st night!
2013 = (3×11×61)
2014= (2×19×53)
2015= (5×13×31)
The next three consecutive sphenic years will be 2665 (5×13×41), 2666 (2×31×43) and 2667 (3×7×127)! No immediate occasson for us to celebrate this Mathamatical Happening!
Further 2022 is a Polite number. Polite numbers are found in abundance. All numbers which can be expressed as a sum of two or more consecutive numbers are known as Polite numbers.
Only powers of two ( 2,4,8,16,32,64,128,...) can not be expressed as a sum of two or more consecutive numbers. And all these numbers ( 2,4,8,16,32,64,128,...) are known as Impolite numbers.
So we will have to wait for 26 more years to see year 2048; which will be the first impolite year after year 1024! Will we see 2048!?
Very impolite if you say that I will not!
Once again Happy New year to all! Love your numbers whatever they may be and enjoy life!
Pankaj Khanna
9424810575
P.S.
Some advanced level questions for CAT aspirants:
In how many ways 2022 can be expressed as
(1) A sum of two perfect squares?
( 2) A sum of two perfect cubes?
(3) A difference of two perfect squares?
(4) A sum of two perfect cubes?
(5) A sum of several Consecutive Natural numbers?
Now a few questions on language part!
(1) What does the word 'Sphenic' mean ?
(2) What is the origin of the word Sphenic?
(3) What does Sphenoid mean?
You sure have suuficient material inside your Sphenoid and other bones to answer the above questions.
