Charles Babbage (b. 1791)
Babbage was an English mathematician and inventor/engineer. He invented railway
signals and the speedometer, but most importantly he invented a calculating
machine called the difference engine.
The difference engine could find the value of any polynomial function by using
differences and repeated addition (which was easier to do than multiplication).
Polynomials were important as they were used to approximate logarithms and
trigonometric functions, which were used widely in engineering.
How did it calculate the value of a given polynomial function using differences and
addition? Let’s use the function 𝑓(𝑥) = 𝑥 2 + 4 as an example.
The user would start by inputting the first few values of the function and their
differences:
As this is a quadratic, all the 2nd differences will be the same (= 2). So the machine
would use this fact to generate all the rest of the terms in the sequence like this:
So values of the function are generated using only differences and addition.
Try generating different quadratic functions in this way. Can you find a connection
between the differences and the coefficients of the functions?
Generate the first few values of the function 𝑓(𝑥) = 𝑥 2 + 𝑥 + 41.
What can you say about the numbers it generates? Will this always be true? When
does it stop producing these numbers? Why?
Can you find any other interesting functions?
Notes on Babbage
The function 𝑓(𝑥) = 𝑥 2 + 𝑥 + 41 was discovered by Euler. It is interesting because it
appears to generate prime numbers. The first few numbers it generates are 41 (if
you include x=0), 43, 47, 53, 61, …
Of course, it won’t generate primes forever as f(41) can’t be prime. Goldbach proved
that there is no polynomial with integer coefficients that will generate primes
forever.
In fact it generates primes until f(40) = 1681 which is 412 . You can see why 402 +
40 + 41 = 412 , because 40(40 + 1) = 412 − 41 = 40 × 41
So this polynomial generates 40 primes. We can use it to create a function that
generates 80 primes as follows:
𝑓(𝑥) = (𝑥 − 40)2 + (𝑥 − 40) + 41
Other quadratics that generate lots of primes are 𝑓(𝑥) = 𝑥 2 + 𝑥 + 17 and 𝑓(𝑥) =
2𝑥 2 + 29.
Can you find quadratics that generate no primes?
We can find the values this function generates in this number spiral:
Can you explain why the numbers appear at the corners of this spiral?