Year of Hope 2021 – The Mathematical Views

2020 is about to pass and 2021 is coming. This is the moment of leaving the old to welcome the new. Wishing you all a Happy New Year! May all your dreams and wishes come true.

Of course, 2020 was a year of challenge for all of us. Lives were lost unnecessarily to Covid 19. The year was lost for students, families, weddings, holidays, positive human interaction.

But in 2021, we all are hopeful about reopening of schools and other educational institutes, vaccine efficacy, great incomes, recovery of lost relationships and no more loss of lives due to the disease.

Despite of all these, let’s see a mathematical hope in 2021. To begin with, in Roman Numerals 2021 is represented by MMXXI.

The number 2021 is composite, which is the product of two consecutive primes 43, 47. It is the concatenation of two consecutive integers 20 and 21.

There are four divisors of 2021, namely 1, 43, 47, 2021 and sum of the divisors is 2112 (which is larger than the number itself). Hence 2021 is not a perfect number.

Tough the number 2021 is not a Palindrome, but the sum of this number and its reverse (1202) is 3223, which is a Palindrome. Even their product 2429242 is also a palindrome!

The number is odd but it is neither a perfect square nor a perfect cube. We can write 2021 as 20 + 21 + 22 +… + 65 + 66.

The first 33 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131 and 137. The sum of 33 and these 33 primes is certainly 2021!

A domino is an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which (i, j) and (j, i) do not both appear for any i and j. Consider all the prime numbers below 100, i.e., 2, 3, …, 89, 97 and make domino pairs: (2, 3), (3, 5), (5, 7), …, (83, 89), (89, 97). The sum of the numbers in all the pairs is 2 + 3 + 3 + 5 + 5 + 7 + … + 83 + 89 + 89 + 97 = 2021.

Let’s place any of the four mathematical symbols +, –, x, ÷ and parentheses between the numbers, so that the value of the expression is 2021. Below are few examples.

2021 = [10 x (9 + 8 − 7) + (6 −5)] x [4 x (3 + 2)] + 1

2021 = [10 x 5 – 3] x (7 x 6 + 1) ÷ [8 +4 – (9 + 2)]

2021 = 10 – 9 x 8 x 7 x 6 + 5 – 4 x 3 x 2 – 1

2021 = (10 + 9) x 8 x (7 + 6) + 5 x (4 + 3 x 2 −1)

2021 = 10 x (− 9 + 8 + 7 x 6 + 5 x 4 x 3) x 2 + 1

If we allow fractions and various operations associated with it, then

2021= ([{(10 + 9) x (8 + 7 / 6)} – 5] x 4 – 3) x (2 + 1)

Finally using exponentiation, 2021 can be achieved in the following ways:

2021 = 10 + (9 + 8 + 7 + 6)(5 + 43 – 2) + 1

2021 = (1+1)11 − (1+1+1)(1+1+1)

 

Once again, “Happy New Year 2021!”. Stay healthy and safe.

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