There is a line of research into the prime sequence that has not yet, to my knowledge, been done so thus far.
An arithmetical result is that an uneven composite integer having two integer factors can be expressed as the difference of two squares of integers, one of the squares being even, the other square uneven. The prime factors are then the two integers that have been squared. If these two squared numbers can be found then the factors are known. It is very easy to make up examples of this.
In an analogous manner, if a prime number can be expressed as the sum of two squares, then this immediately factors into a complex number and its conjugate, where the real and imaginary parts are integers. If this can be done for a series of prime numbers then it may just transpire that the double series of real and imaginary parts may be predictable, and the successor primes directly computed. Elementary logic and Diophantine equations might be involved. For example, an algebraic expression has to represent an even integer if there is an even integer on the other side of the equality sign.
The sequence of the primes has remained an unsolved problem since Euclid C. 2500BC so the student should not feel downhearted if he or she has not managed to resolve this perplexity. According to Georg F. B. Riemann, the eminent 19th century German mathematician, the prime sequence is a matter which the human mind will never be able to understand .
Never the less, the expression of primes as the sums of two integer squares looks promising.
This is not an answer, but a pointer to research, rather than hopeless speculation.
[A foot note] The sequence of primes is conventionally written as: [2, 3, 5, 7, 11, ... ] This is not really logical because the number two [2] functions in arithmetic very differently from the infinitude of the remaining primes: [3, 5, 7, 11, 13, ... ] It might be better to write: [2], [3, 5, 7, 11 ...] so placing two in a class of its own.]
Judging by the inventiveness and imagination of the individuals who designed, built and programmed this computer on which I am now typing, I can see no practical limit to man's ingenuity except the limit of his/her imagination.
Answering my own question - In the first place, there is so much injustice on earth - that has to be righted elsewhere. Man/woman is created for happiness. An optimum amount of wealth produces security and staves off unhappiness and even brings contentment but perfect happiness requires an absence of unhappy memories and a perfect balance of the dualism - the fulfillment of one's own just desires and the fulfillment of one's obligations to society. Each day doing everything required of one does not necessarily give pleasure, but it certainly promotes happiness. The best antidote to depression is [since I speak from a Christian background] to seek to do God's will every day - with his help and just do that. Perfect happiness in the Christian viewpoint requires amongst other things permanence [eternity], a sense of deservedness, a unity of spirit with others and ultimately is the vision of God. Saint Thomas Aquinas wrote down all about happiness far better than I could ever express this matter in his Summae Theologica. There is an amusing quote from the late Ingrid Bergman: " Happiness is having good health and a bad memory ". Death ensures the deletion of all bad memories. In Heaven - the goal of all human life - everything will be present for perfect happiness. There is another place - the very antithesis of Heaven - the Bible warns us all about that dreadful place and tells us how to avoid it. The Bible also tells us that anyone with even the least humility - a moral virtue - would never go to Hell. Once, in casual conversation, someone wished that a well known politician should " burn in Hell ". I thought what an impious thought, and said a prayer for that person and the politician - that this scenario would never happen.