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Reviewer: Albert Alkins Mullin

G. H. Hardy believed that the supreme beauty of number theory resided in the crisp rigor of its logic and the paucity of its practical applications. Surely the present book, which emphasizes the vigor and intuition of number theory, is a counterexample on both counts. The author uses a discursive and intuitive style, rather than the more conventional theorem-proof style. Contradictory as it may sound, this book is actually about intuitive number theory and its applications to cybernetics. To this extent, it uses a rare approach that H. Minkowski and A. M. Turing would have approved of. Indeed, the more intuitive the number theory, the more applicable it is, if true] Primary emphasis is given to elementary number theory and its applications, although formal analytic number theory is not ignored altogether. Among the numerous topics discussed are the following: Fibonacci numbers, cyclotomy; Fermat's Last Theorem (including one misleading sentence about G. Falting's important corollary to his positive solution of Mordell's 1922 Conjecture [1]); the failure of the general converse to Fermat's Theorem; Euler's function and its application to public-key cryptosystems; Galois Fields (finite fields) and their applications to coding theory; unique factorization; formulas for primes (but no mention of Matijasevic?'s integral polynomials [2] whose positive ranges represent all and only primes); the distribution of primes; the Riemann zeta-function and large nuclei, twin primes, squared squares, and electric circuits; diophantine equations in physics; primitive roots and their Fourier property; knapsack encryption; quadratic residues; spread spectrum techniques; fast Fourier transforms; coin tossing by telephone; Dirichlet series; partitions; cyclotomic polynomials; concert hall acoustics; active antenna arrays; and Gaussian primes. The book is very well written, but one must still wonder why so few of the classical number-theoretic ideas are not intuitively transformed; e.g., intuitively one could have applied the Fundamental Theorem to itself to obtain new infinite families of number-theoretic objects which generalize many classical number-theoretic objects and specialize others. Further, since most Fermat numbers are composite, how can they be classified and characterized__?__ Here one can prove that F 5, F0I6, F 6, F 7, F 8, are the only four consecutive Fermat numbers that are products of two distinct primes. Similar research could be done on Mersenne composites, too] Finally, it is unfortunate that the author did not devote some space to intuitive offsprings of Faltings' Theorem. In less than a page, he could have shown that Fermat's last Theorem is valid for at least 95 percent of all exponents n. In only slightly more space, he could have shown that Fermat's Last Theorem fails on a set of exponents n sparser than the primes. However, computer scientists may be saddened, since both proofs are noneffective, just as Go¨del's Completeness Theorem for First-Order Logic is noneffective (see [3]). The author has successfully pushed Intuitive Number Theory to its logical conclusions, but hasn't come close to pushing it to its intuitive conclusions. I could find only two candidate errors. On p. 17, the philosophical quotation is probably due to Dedekind rather than Gauss; on p. 39, delete n=2, since there cannot be any polygon for this case.