Of math and coffee: the history of 124, the untouchable number

"Mathematicians are machines that turn coffee into theorems". The phrase is from Alfred Renyi, one of the "Martians", as Enrico Fermi called them: Hungarian scientists who, with the advent of Nazism, fled to the United States before the Second World War.

So, before reading this post, grab a nice bottle of Nº 124. Drink it to the last drop. Then read and send us any other property of the number 124 that we may have overlooked ....

As everyone knows, Nº 124 "stole" the 124 from Tom Tjaarda's artpiece, the fantastic 124 Spider for Pininfarina. Tjaarda, an American designer of Dutch origins, designed about forty Italian cars, including the Pantera De Tomaso, the Lancia Thema and the FIAT Barchetta. He was rightly called "the master of proportions".

Proportions lead back to mathematics. Pythagoras thought that the universe could be described through the use of proportions –hence the name Kosmos, order. Now we know that rational numbers, so dear to Pythagoras, are an infinitesimal of the irrational ones - that is, given two lengths measured in the real world it is impossible for the relationship between the two to be a rational number.

Not for this proportions, rational or not, are not important. And proportions are ratios between numbers, and 124 is a number.

A question therefore comes to our mind :) –what kind of number is it?

• Obviously it is an even number. It also differs from a power of 2 to another power of two (4 is missing to get to 128)
• It is non-totient. That is, there is no number that has 124 numbers, smaller than the number itself, coprimes (two numbers are coprime if they have no common divisors except 1)
• It is the 17th non-totient even number
• The sum of the digits is a prime number (7), which is also the power of 2 it approaches (128 = 2 to the power of 7)
• It is an untouchable number, i.e. there is no number whose divisors (including 1), added together, give 124.
• It is the sum of 8 consecutive prime numbers (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29)
• It goes without saying that, being untouchable, it is not perfect: the sum of the divisors of perfect numbers gives in fact the number itself (e.g. 28 = 1 + 2 + 4 + 7 + 14). This does not happen for 124.

But happily, even if the number 124 is imperfect...  the drink is (or at least we try)!

Thanks to Adolfo Zilli: the 3 digits are consecutive power of two. 1=2^0, 2=2^1, 4=2^2. Such an evident property was overlooked!

But there is more: the binary representation of 124 in a 3x3 matrix is:

0 0 1
0 1 0
1 0 0

Another friend, Matteo Gallone, immediately squared the matrix and found... the identity matrix.