Looking at D.E. Littlewood's Congruences and Prime Modulus

 

As I noted in my March 13 post, math legend D.E. Littlewood also treated congruences via the Euclidean Algorithm in his timeless monograph 'The Skeleton Key Of Mathematics.

In this post we want to see how Littlewood arrives at congruences. We consider moving around the clock face with the hour hand going beyond the 12.  It is clear looking at the clock that to go forward four hours or to go forward sixteen hours amounts to the same thing, in so far as the final position of the clock's hands are concerned.  We can say, after Littlewood (p. 30):

"Four is congruent to sixteen to modulus twelve"

Which is written:

   4   ≡   16 (mod 12)

As Littlewood puts it:  

"In general p is congruent to q (mod n) if and only if  (p - q) is divisible by n"

So we check to see for this case, given p = 4, q = 12 so that (p - q) = 4 - 12 = -8

Then (p - q)/ n =  -8/ 12 =  - 2/3

Thus, Littlewood concludes:

"Every integer p is congruent (mod n) to one of the numbers: 0, 1, 2,...., (n- 1). The particular number is found by dividing p by n until the reminder is less than n and considering this remainder, which is called the residue  (mod n)."

Adding: "The most interesting case is when the modulus is a prime number."

He calls this number a "prime modulus"

Littlewood goes on to elaborate the number system he is proposing:

"We define the product and the sum of two residues (mod p) to be the residue to which the product or sum of the numbers is congruent. In this way we obtain a number system which has many of the properties of the ordinary integers.  Thus addition and multiplication are defined and these obey the commutative, associative and distributive laws."

One analogous (not exactly alike) simple system I showed in a 2010 post to do with group theory. This specifically applied to 'clock groups' which bear a similarity to Littlewood's modular 12-hour clock. One example is the clock group defined in the image below:

 As we can see there are four members, 0, 1, 2, and 3. The process of addition is defined by adding elements – starting with 0- in a clockwise sense. Doing this we should be able to find a complete closed set of addition operations for all the elements. For example, we find 0 + 1 =1, and 0 + 2 = 2 and so forth. Similarly, we find 1 + 1 = 2, 1 + 3 = 0, 2 + 3 = 1 and so on. Each result obtained by adding the portion of the cycle from the starting element. From here, we may set out the group under addition (+) (Fig. 1- right top).

In the case of multiplication we can write, for example:

1 x 2 = 2 (mod 4)  or:   2 x 2 = 0 (mod 4)  or: 3 x 2 = 2 (mod 4)

Note, however, this system is not the same as the one Littlewood is proposing. This is because the commutative and distributive laws are not consistently obeyed by groups.

Note also the examples given above are nothing to do with Littlewood's prime modulus case because n = 4 is not a prime number.

Littlewood makes the case that in one respect the residues to a prime modulus resemble the rational numbers rather than the integers (i.e. division is always possible except by zero.).

Consider Littlewood's example (to modulus 7):

Since: 4 x 2 = 8  ≡ 1 (mod 7)

Then we can also write:

1   ÷  2 = 4 (mod 7) 

And:

1   ÷   4 = 2 (mod 7) 

Division by 7 is obviously impossible since 7  ≡   0  (mod 7) which is equivalent to division by zero.

For the general prime modulus p, Littlewood notes:

"Division is accomplished by the use of Euclid's Algorithm, which suffices to prove that division except by 0 is always possible."

Suggested Problems:

1)  Write four additional equivalences for the mod 7 residues (2 already shown)

2)  Consider the p = 5 prime modulus portrayed as shown below in clock form:

Write four equivalences for the mod 5 residues. Can these be written in the form for the mod 7 cases in Problem 1? Why or why not?