2021

With 2021 almost upon us (I'm writing this on December 31st 2020), it's appropriate to look at some of the mathematical properties associated with this number. I'm quoting from Numbers Aplenty unless otherwise stated.

FIRST FACT 

2021 is rather special: it is the concatenation of two consecutive integers (20 and 21) and also the product of two consecutive primes (43 and 47). In other words:$$2021=\underbrace{2 0 2 1}_{\text{concatenate 20 and 21}}=43 \times 47$$

The next such number is:  

23073409469011482307340946901147 which is the product of the primes 4803478892324963 and 4803478892324969. 

A further example, with the two parts in increasing order like 2021, is given by the concatenation of:  

794018604377235322848433897872605582 and 794018604377235322848433897872605583.

Unsurprisingly, SageMathCell fails to factorise the catcatenated number

SECOND FACT

Let's take all the prime numbers below 100, i.e., 2, 3, 5, 7, 11, ..., 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 2021. In other words:$$2021 = \underbrace{2 + 3}+ \underbrace{3 + 5} + \underbrace{5 + 7} + ... + \underbrace{83 + 89} + \underbrace{89 + 97}$$

THIRD FACT

2021 is also equal to 33 plus the sum of the first 33 primes. In other words:$$2021 = 33 + \underbrace{2 + 3 + ... + 127 + 131 + 137}_{ \text{first 33 primes}}$$

FOURTH FACT

It is a junction number, because it is equal to \(n\)+sod(\(n\)) for \(n \)= 1996 and 2014. In other words:$$2021 = 1996 + \underbrace{25}_{\text{sod}} = 2014 + \underbrace{7}_{\text{sod}}$$

FIFTH FACT

2021 is a member of Euler's famous prime generating polynomial \(n^2+n+41\) for the case where \(n\)=44. The output in this case of course is not a prime but a semiprime. These numbers form OEIS A202018. In other words:$$2021=\underbrace{44^2+44+41}_{n^2+n+41 \text{ when }n=44}$$ SIXTH FACT 

 2021 is an emirpimes, since its reverse is a distinct semiprime:$$\underbrace{1202}_{\text{reverse }2021} = 2 \times 601$$