Tweet

Scott Kominers

1 Jan, 18 tweets, 5 min read

Happy new year! Interesting numerical properties of 2021: a thread👇

2021 = 4347 is semi-prime, meaning it's the product of two primes – and that's not all: it's a product of "cousin primes" that differ by only 4. The last such year was 1517, and the next is 4757.

Even the next product of consecutive primes is a long way off – that's 2491.

Plus 2021 is a prime in base 3 and base factorial!

And 2021 yields a prime whenever you insert a "0" between any two of its digits (which last happened in 1909).

2021 is also the sum of the first 24 pairs of consecutive primes.

The 2021st pentanacci number is prime, as is (2^2021 + 2021 + 1)2^2021 + 1.

2021 is also the number of prime juggling patterns of period 11 using two balls. Period 12 has 4146.

@2600

Meanwhile, the 2021st prime plus 2021 is a square. That last occurred in 1334, and happens next in 2455 and then the excellent 2600. (cc @2600)

And 2021 appears in the Fibonacci sequence beginning with 13 and 6; it's followed by 3270 and then 5291.

Moreover, 2021 appears as a permutation/combination sum:

2021 = Sum_{k=0,...,6}[Permutation(6,k)+Combination(6,k)].

Replacing 6 with 7 gives 13828.

2021 is the number of totient-partitions of 49. Of course, 48 has 42 totient-partitions; 51 has 17355.

2021 also appears as a denominator of a coefficient in a particular continued fraction expansion of the Gamma function. The next one is 125896643.

And thinking about the cross-year transition: (2020)^1024 + (2021)^1024 is prime (which happens next in 2471).

2020 + 2021 = 4041 is a Shift-Moebius transform of a power of 2 (the last one was 1987, and the next is 8071).

And the 4041st triangular number contains only the digits 1, 6, and 8 (the next triangular number with this property is the 36908th).