1. Introduction


This paper describes seven unsolved problems in sequence algebra encountered by the author mostly from previous publications [3 to 11].


(a) Unsolved Problem 1: Direct formulation for Prime(z)


The author succeeded in deriving a direct global formulation for the composite number sequence given by equation (1) [ 9]:


...............................................................1
.......................................Comp(z) = ------------- .............................................(1).
...........................................................i...i
.........................................................z..(z..-.1)

The development was based on the historical Sieve of Erastothenes with a twist [1]. Instead of attacking the prime sequence directly, the author chose to attack the composite sequence. This turned out to be much easier as evidenced by the simple formulation.

From the sequence identity given in equation (2), one hopes to find the global prime sequence given by equation (3). Note that equation (3) does not involve any primality test and the predictions are determinstic.

...............Nat(z) = Normc(Comp(z)) + Prime(z) + 1 + 1/z ................(2)

This leads to


................Prime(z) = Nat(z) - Normc(Comp(z)) - 1 - 1/z ...................(3).

Comp(z) in equation (1) is called an analytical sequence whereas Prime(z) in equation (3) is called an algorithmic sequence. If sequences are in analytical format, one could continue to manipulate and build higher analytical sequences. On the other hand if a sequence is algorithmic, one could still develop efficient computer programs using it but continued analysis cannot proceed without the use of the operator Normc( ). Normc( ) itself is algorithmic rather than analytical. In order to solve this problem, one should find a direct sequence formulation for Prime(z). Of course one could write down Prime(z) if a primality test is admitted as shown in equation (4).

Prime(z)=sum(1/z^i*(isprime(i)-false)/(true-false),i=2..100);

........................1......1.........1........1.......1.......1.......1.......1.......1.......1.......1.......1.......1.......1
...Prime(z) := ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ------
.........................2......3.........5........7......11.....13.....17......19.......23.....29.....31.....37.....41.....43
........................z......z.........z........z.........z.......z.......z.........z.........z.........z.......z.......z.......z.......z

............1......1.......1........1.......1.......1.......1.......1.......1.......1.......1
.......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ------ .......................(4).
.............47.....53....59....61.....67.....71.....73.....79.....83.....89.....97
............z......z.......z........z........z.......z.........z.......z.......z.......z.......z

(Abstracted from Maple V R 3's help page for Isprime: The function isprime is a probabilistic primality testing routine. It returns false if n is shown to be composite within one strong pseudo- primality test and one Lucas test and returns true otherwise. If isprime returns true, n is ``very probably'' prime - see Knuth ``The art of computer programming'', Vol 2, 2nd edition, Section 4.5.4, Algorithm P for a reference and H. Reisel, ``Prime numbers and computer methods for factorization''. No counter example is known and it has been conjectured that such a counter example must be hundreds of digits long.)

Primality tests are probabilistic which are efficient but are algorithmic formulations which cannot be embedded in analytical formulations for further manipulations. This is because such tests return logical switches between true and false which function like mixed mode arithmetics in sequence algebra.