## 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 2^{3} 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 (2222^{2} = 4,937,284 = 192^{2}+193^{2}+...+279^{2}).

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

## 22222...

22222 is a Kaprekar number because 22222^{2} = 493,817,284 and 4,938+17,284 = 22,222.

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