#Proof in a Tweet

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The first tweet shown here by @republicofmath launched the idea of a new math hashtag : #proofinatweet. Here is a collection as a "Moment" on Twitter :
Many thanks to all the contributors.

- Below are some of the tweets I managed to save before they were forever lost in the Twittosphere. [edit :] That was in 2012 before Twitter invented the "Moment" thing.

Alexander Bogomolny ‏@CutTheKnotMath
ΔABC, C is right angle. Altitude CH makes 2 Δs similar to ABC. Area of Δ is proportional to square of side ∴ a²+b²=c² #proofinatweet

Vincent PANTALONI ‏@panlepan
@republicofmath There are arbitrarily long gaps without prime numbers because for all n there are no primes in [n !+2, n !+ n] #proofinatweet

Republic of Math ‏@republicofmath
Polyominoes have even perimeters : when we add a new square previous perimeter edges cancel in pairs #proofinatweet

Vincent PANTALONI ‏@panlepan (reply)
@republicofmath Then, a Polyiamond with n Δ has a perimeter P_n= n mod 2 because P_1= 3 and not 4 and a Δ has 3 sides . #proofinatweet

K P Hart ‏@hartkp
@panlepan Let f map X to P(X). Then x:x not in f(x) not in range of f. No map from set to power set is surjective #proofinatweet

Vincent PANTALONI ‏@panlepan
If f’=f and f(0)<>0 then f has no root : set g(x)= f(x)f(-x). g’(x)= f’(x)f(-x)- f(x)f’(-x)=0 becos f’=f. So g=cst=g(0)=f(0)^2 proofinatweet

K P Hart ‏@hartkp
.@panlepan (a-b)^2>=0 ; add 4ab, then (a+b)^2>=4ab or a+b>=2sqrt(ab) ; the AGM-inequality #proofinatweet

Tim Kindberg ‏@timkindberg
If real numbers countable then list them. Form r that differs in i’th decimal from i’th real. Not in list ! Reals uncountable #proofinatweet

Vincent PANTALONI ‏@panlepan
There’s a rational number between any two real numbers x a-a’= 0= b-b’ #proofinatweet

Vincent PANTALONI ‏@panlepan
@republicofmath Let x=0.999... then 10x=9.999... so 10x-x= 9 and 9x= 9 so x=1. Hence 0.9999...= 1 #proofinatweet

Vincent PANTALONI ‏@panlepan
@MarcusduSautoy Suppose there are n primes. Then prod(p_i)+ 1 can’t be divided by any of the p_i so is a prime. Absurd.