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On bounded arithmetic augmented by the ability to count certain sets of primes. (English) Zbl 1174.03026

The paper is concerned with the problem of proving the existence of infinitely many primes in variants of bounded arithmetic. J. B. Paris, A. J. Wilkie, and A. R. Woods [J. Symb. Log. 53, No. 4, 1235–1244 (1988; Zbl 0688.03042)] proved infinitude of primes in the theory \(\text{I}\Delta_0+\Omega_1\), but it does not seem to be provable in \(\text{I}\Delta_0\) alone. A. R. Woods [Some problems in logic and number theory, and their connections. Ph.D. thesis, Univ. Manchester (1981)] conjectured that the infinitude of primes is provable in the theory \(\text{I}\Delta_0(\pi)\) augmented by a recursive definition of the prime-counting function \(\pi\).
In the present paper, the authors prove the existence of infinitely many primes, and, in fact, Bertrand’s postulate (stating that there is a prime in the interval \((x,2x]\) for every \(x>0\)), in a similar but somewhat stronger theory, \(\text{I}\Delta_0(\xi)+\text{def}(\xi)\), where \(\text{def}(\xi)\) denotes a natural recursive definition of the function \(\xi(x,y,e)\) counting the number of primes \(p\leq x\) such that \(\lfloor y/p^e\rfloor\) is odd. Since the proof heavily relies on counting with logarithms, the paper includes a detailed development of a suitable rational approximation to the natural logarithm function, as well as several prime-counting functions, such as Chebyshev’s \(\theta\) and \(\psi\), in bounded arithmetic.

MSC:

03F30 First-order arithmetic and fragments
03F20 Complexity of proofs
11A41 Primes

Citations:

Zbl 0688.03042
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References:

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