In this question arithmetic in a restricted subset of Z, similar to arithmetic with E-numbers,

is investigated. Let D = {4a + 1 | a ∈ Z} and call the elements of D the *D-numbers*. The

first few positive D-numbers are

1

, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57.

Put another way, the D-numbers are the integers which are congruent to 1 modulo 4. Use

of arithmetic modulo 4 makes answers to some of the the following questions very simple

(but is not mandatory).

(a) Show that if two D-numbers are multiplied together the result is a D-number. Give an

example to show that the same is not true when two D-numbers are added together.

(b) If a and b are D-numbers we say that a D-divides b if b = ac, where c is a D-number.

Show that 5 D-divides 25 and 45. Show that 1 D-divides every D-number.

(c) Now show that if a and b are D-numbers such that a|b (in the usual sense of Definition

1.5) then a D-divides b. Give an example to show that there are integers n and m,

which are not both D-numbers but are such that mn is a D-number.

(d) If a is a positive D-number greater than 1 and the only positive D-divisors of a are

1

and a , then we say that a is *D-prime*. List the first 10 D-primes and the first two

positive D-numbers (> 1) which are D-composite (i.e. not D-prime).

(e) All (ordinary) odd primes are either of the form 4m+1 or 4m+3: that is are congruent

to 1 or 3 modulo 4. Show that a D-number is D-prime if and only if it is prime (in Z)

or its prime factorisation is pq where p and q are congruent to 3 mod 4 (and may be

equal).

(f) Find a D-number which has two distinct D-prime factorisations.