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	Two Series containing the Primes and a
	Conjecture concerning the Prime Difference
	A constructive attitude towards prime Numbers
	B. S. Anand (28/08/1964)
I	Hardy and Wright ask some natural questions regarding primes in their
	introductory book on number theory. Which suggest the related enquiry
	as to the smallest prime greater than a given positive integer x.
	A constructive reply is easily obtained.
	I consider that multiple of p - a prime smaller than x - which immediately
	exceeds x.
	The smallest odd number not amongst such multiples is the required prime.
	With minor refinements, this result may be used to construct the prime
	series. As distinct from the sieve of Erastosthenes where the primes are
	discovered amongst the series of integers. A construction which I now
	proceed to develop formally. And which may have some significance in an
	age of computers.
II	The primes smaller than x I take to be p(i) for i = 1 to n. With p(1) = 2.
	And p(n) as the n'th prime, immediately smaller than x.
	I have the congruences:
	x + a(i) ≡ 0 (mod p(i)) ... 0 ≤ a(i) < p(i)
	I can form the class {a(i)} of residues a(i) mod p(i). Any member of this,
	when added to x, gives an integer divisible by p(i).
	But what interests me more are the integers not divisible by p(i). So I form
	the complement of {a(i)} in the non-negative integers. And denote it by
	{a(i)}*. It is non-empty as it contains a(i) + 1.
	Now, a common member of these classes {a(i)}*, when added to x,
	yields an integer not divisible by any p(i). The solutions y of the n
	simultaneous congruences:
	y ≡ a(i) + 1 (mod p(i))
	being such common elements, ∩{a(i)}* is not empty.
	I denote its smallest member by y(m).
	In case x is composite, this gives x + y(m) as the smallest prime
	greater than x.
III	There is a useful result in connection with primes. That only the primes
	smaller than the square root of x are necessary to test its primality.
	So I am suggested the theorem:
	For every w satisfying:
	(i) w is in ∩{a(i)}*, w - y(m) < (x + y(m))*(x + y(m-1)), if x is composite;
	(ii) w is in ∩{a(i), b}*, w < x*(x = 1), if x is prime;
	the integer x + w is a prime.
IV	The reasoning in I and II may be graphically presented by means of the
	attached prime number construction charts, where I work directly with
	the various a(i)'s, instead of their residue classes, which results in the
	occurence of some composite numbers, as indicated.
	I illustrate the method of construction by considering, say, the k'th row which
	has no zeros.
			p(1)	p(2)	...	p(i)	...
			...	...	...	...	...
			...	...	...	...	...
	k	p(n)	a(1, k)	a(2, k)	...	a(i, k)	...
	Here, for i < n, 0 < a(i, k) < p(i):
	(i)	k is the number of the construction;
	(ii)	n is the number of the prime given by the (k-1)th step;
	(iii)	a(i, k) is given by p(n) + a(i, k) ≡ 0 (mod p(i));
	(iv)	r(n) is the least even integer not among the a(i, k)'s.
	If p(n) + r(n) is not congruent to 0, (mod p(i)), for any i ≤ n, it is therefore
	the prime p(n+1).
	To obtain the (k+1)th row, subtract r(n) from a(i, k) to give a(i, k+1), adding p(i)
	if necessary. Subtract it also from p(n) to give a(n, k+1).
	And p(n) + r(n) will not be prime only when r(n), though not equal to any a(i, k),
	nevertheless exists in some residue class {a(i, k)}.
	In which case one of the a(i, k+1) will be zero, and we have generated a composite
	number.
	And every time r(n) = 2, a prime pair occurs.
	While if in any row a(1) = 1, and a(i) is not 0 or 1 for any other i, then the construction
	has generated a prime of the form 2m+1.
	I may now enquire:
	When is a composite generated?
	And can this procedure be used more effectively than the sieve of Eratosthenes
	for the construction of primes by computers?
	And can problems regarding primes be translated into problems about the
	residues a(i) and the differences r(n)?