By Lucius Annaeus Seneca
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D J J 7 2 −2 Try to decrement 7. Go back and repeat. Halt. 2. MOVE from r to s: a program to move a number from register r to register s (where r = s). CLEAR s. D J I J r 3 s −3 Use the program of the first example. Take 1 from r. Halt when zero. Add 1 to s. Repeat. This program has seven instructions altogether. It leaves a zero in register r. 3. ADD 1 to 2 and 3: a program to add register 1 to registers 2 and 3. D J I I J 1 4 2 3 −4 This program leaves a zero in register 1. It is clear how to adapt the program to add register 1 to more (or fewer) than two registers.
Then we have the equation gQ (x) = G(x, CQ (x)). showing that gQ is a general recursive partial function. Now if we also have another k-place general recursive partial function f , and we define h(x) = f (x) if Q(x) g(x) if not Q(x) then h is a general recursive partial function because h(x) = f Q (x) + gQ (x). 24A. Assume that g is a general recursive partial (k + 2)-place function, and let f be the unique (k + 1)-place function for which f (x, y) = g( f (x, y), x, y) for all x and y. ) Then f is also a general recursive partial function.
So any search for a prime larger than x need go no further than x! +1. 42 Computability Theory Digression: There is an interesting result in number theory here. “Bertrand’s postulate” states that for any x > 3, there will always be a prime number p with x < p < 2x − 2. ) In 1845, the French mathematician Joseph Bertrand, using prime number tables, verified this statement for x below three million. Then in 1850, the Russian P. L. Chebyshev (Tchebychef) proved the result in general. In 1932, the Hungarian Paul Erdo˝ s gave a better proof, which can now be found in undergraduate number theory textbooks.