On Twos for Twosday

At 22 seconds past 22 minutes past ten in the evening of the 22nd day of the 2nd month of the 22nd year this century (which falls on a Tuesday, or Twosday), the time and date will consist of a single repeating digit: the last occasion this will happen in our lifetimes.  In the date format dd/mm/yyyy, 22/02/2022 is also a palindromic date, i.e., is read the same backwards as it is forwards.  To mark and celebrate Twosday, here are a few curios about twos... 

2...

2 is the first (and only even) prime.

2 is the only number which, when added to itself, gives the same result as when multiplied by itself (2+2 = 2×2). 

2 is the only difference between two consecutive primes that is prime  (3 and 5, 5 and 7, 11 and 13...).

2 is the smallest prime that produces another prime seven consecutive times by adding a digit (2, 29, 293, 2939, 29399, 293999, 2939999, 29399999).

2 has a square root that is probably the first number known to be irrational, known as Pythagoras' Constant (watch this Numberphile video, and this from VSauce).

2 is the number most feared by primonumerophobics.

2 is strobogrammatic (the same upside-down) in an 8-segment display.

2 has 2 homphones (to and too).

2 is a child's age on 22/2/22 if they were born on 22/2/2020 (children born at 20:20 on 6/12/2019 will be 2 years, 2 months, 2 weeks, 2 days, 2 hours and 2 minutes old at 22:22 on 22/2/22).

Listen to 'Two - At the double' from Simon Singh's 'Numbers'.

Listen to 'Funbers 2' from Tom Rocks Maths Funbers broadcast.

22...

22 is the first self-describing number (it has 'two twos").

22 is the smallest number that can be expressed as the sum of 2 primes in more than 2 ways (3+19 = 5+17 = 11+11).

22 is the number of ways 2³ can be expressed as the sum of positive integers (8, 7+1, 6+2, 6+1+1, 5+3, 5+2+1, 5+1+1+1, 4+4, 4+3+1, 4+2+2, 4+2+1+1, 4+1+1+1+1, 3+3+2, 3+3+1+1, 3+2+2+1, 3+2+1+1+1, 3+1+1+1+1+1, 2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+1+1+1+1+1+1, 1+1+1+1+1+1+1+1). 

22 is the largest known arithmetic sequence of primes, starting with 11,410,337,850,553 and ending with 108,201,410,428,753, with a common difference of 4,609,098,694,200.

22 is the number of different ways five pentagons can be linked together (i.e., there are 22 pentahexes).

22 has a digit sum (2+2 = 4) equal to the sum of the digits of its prime factors (2, 11... 2+1+1).

22 has a digit sum (2+2 = 4) equal to the product of its digits (2×2 = 4).

22 is the maximum number of pieces that can be created when cutting a circle with six lines (i.e., 22 is the 6th term of the lazy caterer's sequence).

22 is a person's age on 22/2/22 if they were born on 22/2/2000 (people born at 00:00 on 06/12/1999 will be 22 years, 2 months, 2 weeks, 2 days, 22 hours and 22 minutes old at 22:22 on 22/2/22).

222...

222 is the product of exactly three distinct primes (2×3×37).

222 is the sum of 2 successive primes (109+113).

222 is the sum of three squares in at least three ways (1+25+196, 1+100+121, 4+49+169).

222 is the sum of all 2-digit primes formed from consecutive digits (23+43+67+89).

222 is the 22nd number that produces a prime when divided by the sum of its digits [222÷(2+2+2) = 37].

222 has factors that can be partitioned into 2 sets with equal sums (1+2+3+222 = 6+37+74+111).

222 cannot be written as a number + the sum of the number's digits, for any number.

2222...

2222 is the smallest number divisible by a 1-digit, 2-digit and 3-digit prime (2222÷2 = 1111, 2222÷11 = 202, 2222÷101 = 22).

2222 has factors that are all palindromic (1, 2, 11, 22, 101, 202, 1111, 2222).

2222 squared is the sum of 88 consecutive squares (2222² = 4,937,284 = 192²+193²+...+279²).

2222 is the sum of six cubes in exactly six ways.

22222...

22222 is a Kaprekar number because 22222² = 493,817,284 and 4,938+17,284 = 22,222.

22222 is the smallest palindromic multiple of 41 with even digits (22222 = 41×542).