Unsolved Problems In Number Theory by Richard Guy

Unsolved Problems In Number Theory

byRichard Guy

Hardcover | July 13, 2004

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Mathematics is kept alive by the appearance of new, unsolved problems. This book provides a steady supply of easily understood, if not easily solved, problems that can be considered in varying depths by mathematicians at all levels of mathematical maturity. This new edition features lists of references to OEIS, Neal Sloane's Online Encyclopedia of Integer Sequences, at the end of several of the sections.

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Title:Unsolved Problems In Number TheoryFormat:HardcoverDimensions:456 pages, 9.25 × 6.1 × 0 inPublished:July 13, 2004Publisher:Springer New YorkLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0387208607

ISBN - 13:9780387208602

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Table of Contents

Preface to the First EditionPreface to the Second EditionPreface to the Third EditionGlossary of SymbolsA. Prime Numbers.A1. Prime values of quadratic functions.A2. Primes connected with factorials.A3. Mersenne primes. Repunits. Fermat numbers. Primes of shape k · 2n + 1. A4. The prime number race.A5. Arithmetic progressions of primes.A6. Consecutive primes in A.P. A7. Cunningham chains.A8. Gaps between primes. Twin primes. A9. Patterns of primes.A10. Gilbreath's conjecture.A11. Increasing and decreasing gaps.A12. Pseudoprimes. Euler pseudoprimes. Strong pseudoprimes. A13. Carmichael numbers.A14. "Good" primes and the prime number graph.A15. Congruent products of consecutive numbers. A16. Gaussian primes. Eisenstein-Jacobi primes. A17. Formulas for primes. A18. The Erd½os-Selfridge classi.cation of primes.  A19. Values of n making n - 2k prime. Odd numbers not of the form ±pa ± 2b. A20. Symmetric and asymmetric primes. B. Divisibility B1. Perfect numbers. B2. Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers. B3. Unitary perfect numbers. B4. Amicable numbers. B5. Quasi-amicable or betrothed numbers. B6. Aliquot sequences. B7. Aliquot cycles. Sociable numbers. B8. Unitary aliquot sequences. B9. Superperfect numbers. B10. Untouchable numbers. B11. Solutions of mó(m) = nó(n). B12. Analogs with d(n), ók(n). B13. Solutions of ó(n) = ó(n + 1). B14. Some irrational series. B15. Solutions of ó(q) + ó(r) = ó(q + r). B16. Powerful numbers. Squarefree numbers. B17. Exponential-perfect numbers B18. Solutions of d(n) = d(n + 1). B19. (m, n + 1) and (m+1, n) with same set of prime factors. The abc-conjecture. B20. Cullen and Woodall numbers. B21. k · 2n + 1 composite for all n. B22. Factorial n as the product of n large factors. B23. Equal products of factorials. B24. The largest set with no member dividing two others. B25. Equal sums of geometric progressions with prime ratios. B26. Densest set with no l pairwise coprime. B27. The number of prime factors of n + k which don't divide n + i, 0 ¡Ü i <_20_k.0a_b28.20_consecutive20_numbers20_with20_distinct20_prime20_factors.20_0a_b29.20_is20_x20_determined20_by20_the20_prime20_divisors20_of20_x20_2b_20_12c_20_x20_2b_20_22c_.20_.20_.2c_20_x20_2b_20_k3f_20_0a_b30.20_a20_small20_set20_whose20_product20_is20_square.20_0a_b31.20_binomial20_coeffcients.20_0a_b32.20_grimm27_27_s20_conjecture.20_0a_b33.20_largest20_divisor20_of20_a20_binomial20_coeffcient.20_0a_b34.20_if20_there27_27_s20_an20_i20_such20_that20_n20_-20_i20_divides20__nk_.20_0a_b35.20_products20_of20_consecutive20_numbers20_with20_the20_same20_prime20_factors.20_0a_b36.20_euler27_27_s20_totient20_function.20_0a_b37.20_does20_c3b6_28_n29_20_properly20_divide20_n20_-20_13f_20_0a_b38.20_solutions20_of20_c3b6_28_m29_20_3d_20_c3b3_28_n29_.20_0a_b39.20_carmichael27_27_s20_conjecture.20_0a_b40.20_gaps20_between20_totatives.20_0a_b41.20_iterations20_of20_c3b6_20_and20_c3b3_.20_0a_b42.20_behavior20_of20_c3b6_28_c3b3_28_n29_29_20_and20_c3b3_28_c3b6_28_n29_29_.20_0a_b43.20_alternating20_sums20_of20_factorials.20_0a_b44.20_sums20_of20_factorials.20_0a_b45.20_euler20_numbers.20_0a_b46.20_the20_largest20_prime20_factor20_of20_n.20_0a_b47.20_when20_does20_2a20_-2b20_divide20_na20_-20_nb3f_20_0a_b48.20_products20_taken20_over20_primes.20_0a_b49.20_smith20_numbers.20_0a_c.20_additive20_number20_theory20_0a_c1.20_goldbach27_27_s20_conjecture.20_0a_c2.20_sums20_of20_consecutive20_primes.20_0a_c3.20_lucky20_numbers.20_0a_c4.20_ulam20_numbers.20_0a_c5.20_sums20_determining20_members20_of20_a20_set.20_0a_c6.20_addition20_chains.20_brauer20_chains.20_hansen20_chains.20_0a_c7.20_the20_money-changing20_problem.20_0a_c8.20_sets20_with20_distinct20_sums20_of20_subsets.20_0a_c9.20_packing20_sums20_of20_pairs.20_0a_c10.20_modular20_di.erence20_sets20_and20_error20_correcting20_codes.20_0a_c11.20_three-subsets20_with20_distinct20_sums.20_0a_c12.20_the20_postage20_stamp20_problem.20_0a_c13.20_the20_corresponding20_modular20_covering20_problem.20_harmonious20_labelling20_of20_graphs.20_c14.20_maximal20_sum-free20_sets.20_0a_c15.20_maximal20_zero-sum-free20_sets.20_0a_c16.20_nonaveraging20_sets.20_nondividing20_sets.20_0a_c17.20_the20_minimum20_overlap20_problem.20_0a_c18.20_the20_n20_queens20_problem.20_0a_c19.20_is20_a20_weakly20_indedendent20_sequence20_the20_.nite20_union20_of20_strongly20_independent20_ones3f_20_0a_c20.20_sums20_of20_squares.20_0a_c21.20_sums20_of20_higher20_powers.20_0a_d.20_diophantine20_equations20_0a_d1.20_sums20_of20_like20_powers.20_euler27_27_s20_conjecture.20_0a_d2.20_the20_fermat20_problem.20_0a_d3.20_figurate20_numbers.20_0a_d4.20_waring27_27_s20_problem.20_sums20_of20_l20_kth20_powers.20_0a_d5.20_sum20_of20_four20_cubes.20_0a_d6.20_an20_elementary20_solution20_of20_x220_3d_20_2y420_-20_1.20_0a_d7.20_sum20_of20_consecutive20_powers20_made20_a20_power.20_0a_d8.20_a20_pyramidal20_diophantine20_equation.20_0a_d9.20_catalan20_conjecture.20_di.erence20_of20_two20_powers.20_0a_d10.20_exponential20_diophantine20_equations.20_0a_d11.20_egyptian20_fractions.20_0a_d12.20_marko.20_numbers.20_0a_d13.20_the20_equation20_xxyy20_3d_20_zz.20_0a_d14.20_ai20_2b_20_bj20_made20_squares.20_0a_d15.20_numbers20_whose20_sums20_in20_pairs20_make20_squares.20_0a_d16.20_triples20_with20_the20_same20_sum20_and20_same20_product.20_0a_d17.20_product20_of20_blocks20_of20_consecutive20_integers20_not20_a20_power.20_0a_d18.20_is20_there20_a20_perfect20_cuboid3f_20_four20_squares20_whose20_sums20_in20_pairs20_are20_square.20_four20_squares20_whose20_differences20_are20_square.20_0a_d19.20_rational20_distances20_from20_the20_corners20_of20_a20_square.20_0a_d20.20_six20_general20_points20_at20_rational20_distances.20_0a_d21.20_triangles20_with20_integer20_edges2c_20_medians20_and20_area.20_0a_d22.20_simplexes20_with20_rational20_contents.20_0a_d23.20_some20_quartic20_equations.20_0a_d24.20_sum20_equals20_product.20_0a_d25.20_equations20_involving20_factorial20_n.20_0a_d26.20_fibonacci20_numbers20_of20_various20_shapes.20_0a_d27.20_congruent20_numbers.20_0a_d28.20_a20_reciprocal20_diophantine20_equation.20_0a_d29.20_diophantine20_m-tuples.20_0a_e.20_sequences20_of20_integers20_0a_e1.20_a20_thin20_sequence20_with20_all20_numbers20_equal20_to20_a20_member20_plus20_a20_prime.20_0a_e2.20_density20_of20_a20_sequence20_with20_l.c.m.20_of20_each20_pair20_less20_than20_x.20_0a_e3.20_density20_of20_integers20_with20_two20_comparable20_divisors.20_e4.20_sequence20_with20_no20_member20_dividing20_the20_product20_of20_r20_others.20_0a_e5.20_sequence20_with20_members20_divisible20_by20_at20_least20_one20_of20_a20_given20_set.20_0a_e6.20_sequence20_with20_sums20_of20_pairs20_not20_members20_of20_a20_given20_sequence.20_0a_e7.20_a20_series20_and20_a20_sequence20_involving20_primes.20_0a_e8.20_sequence20_with20_no20_sum20_of20_a20_pair20_a20_square.20_0a_e9.20_partitioning20_the20_integers20_into20_classes20_with20_numerous20_sums20_of20_pairs.20_0a_e10.20_theorem20_of20_van20_der20_waerden.20_szemerc2b4_edi27_27_s20_theorem.20_partitioning20_the20_integers20_into20_classes3b_20_at20_least20_one20_contains20_an20_a.p.20_0a_e11.20_schur27_27_s20_problem.20_partitioning20_integers20_into20_sum-free20_classes.20_0a_e12.20_the20_modular20_version20_of20_schur27_27_s20_problem.20_0a_e13.20_partitioning20_into20_strongly20_sum-free20_classes.20_0a_e14.20_rado27_27_s20_generalizations20_of20_van20_der20_waerden27_27_s20_and20_schur27_27_s20_problems.20_0a_e15.20_a20_recursion20_of20_gobel.20_0a_e16.20_the20_3x20_2b_20_120_problem.20_0a_e17.20_permutation20_sequences.20_0a_e18.20_mahler27_27_s20_z-numbers.20_0a_e19.20_are20_the20_integer20_parts20_of20_the20_powers20_of20_a20_fraction20_in.nitely20_often20_prime3f_20_0a_e20.20_davenport-schinzel20_sequences.20_0a_e21.20_thue-morse20_sequences.20_0a_e22.20_cycles20_and20_sequences20_containing20_all20_permutations20_as20_subsequences.20_0a_e23.20_covering20_the20_integers20_with20_a.p.s.20_0a_e24.20_irrationality20_sequences.20_0a_e25.20_golomb27_27_s20_self-histogramming20_sequence.20_0a_e26.20_epstein27_27_s20_put-or-take-a-square20_game.20_0a_e27.20_max20_and20_mex20_sequences.20_0a_e28.20_b2-sequences.20_mian-chowla20_sequences.20_0a_e29.20_sequence20_with20_sums20_and20_products20_all20_in20_one20_of20_two20_classes.20_0a_e30.20_macmahon27_27_s20_prime20_numbers20_of20_measurement.20_0a_e31.20_three20_sequences20_of20_hofstadter.20_0a_e32.20_b2-sequences20_from20_the20_greedy20_algorithm.20_0a_e33.20_sequences20_containing20_no20_monotone20_a.p.s.20_0a_e34.20_happy20_numbers.20_0a_e35.20_the20_kimberling20_shuffle.20_0a_e36.20_klarner-rado20_sequences.20_0a_e37.20_mousetrap.20_0a_e38.20_odd20_sequences20_0a_f.20_none20_of20_the20_above20_0a_f1.20_gauc39f_27_27_s20_lattice20_point20_problem.20_0a_f2.20_lattice20_points20_with20_distinct20_distances.20_0a_f3.20_lattice20_points2c_20_no20_four20_on20_a20_circle.20_0a_f4.20_the20_no-three-in-line20_problem.20_f5.20_quadratic20_residues.20_schur27_27_s20_conjecture.20_0a_f6.20_patterns20_of20_quadratic20_residues.20_0a_f7.20_a20_cubic20_analog20_of20_a20_bhaskara20_equation.20_0a_f8.20_quadratic20_residues20_whose20_di.erences20_are20_quadratic20_residues.20_0a_f9.20_primitive20_roots20_0a_f10.20_residues20_of20_powers20_of20_two.20_0a_f11.20_distribution20_of20_residues20_of20_factorials.20_0a_f12.20_how20_often20_are20_a20_number20_and20_its20_inverse20_of20_opposite20_parity3f_20_0a_f13.20_covering20_systems20_of20_congruences.20_0a_f14.20_exact20_covering20_systems.20_0a_f15.20_a20_problem20_of20_r.20_l.20_graham.20_0a_f16.20_products20_of20_small20_prime20_powers20_dividing20_n.20_0a_f17.20_series20_associated20_with20_the20_c3a6_-function.20_0a_f18.20_size20_of20_the20_set20_of20_sums20_and20_products20_of20_a20_set.20_0a_f19.20_partitions20_into20_distinct20_primes20_with20_maximum20_product.20_0a_f20.20_continued20_fractions.20_0a_f21.20_all20_partial20_quotients20_one20_or20_two.20_0a_f22.20_algebraic20_numbers20_with20_unbounded20_partial20_quotients.20_0a_f23.20_small20_differences20_between20_powers20_of20_220_and20_3.20_0a_f24.20_some20_decimal20_digital20_problems.20_0a_f25.20_the20_persistence20_of20_a20_number.20_0a_f26.20_expressing20_numbers20_using20_just20_ones.20_0a_f27.20_mahler27_27_s20_generalization20_of20_farey20_series.20_0a_f28.20_a20_determinant20_of20_value20_one.20_0a_f29.20_two20_congruences2c_20_one20_of20_which20_is20_always20_solvable.20_0a_f30.20_a20_polynomial20_whose20_sums20_of20_pairs20_of20_values20_are20_all20_distinct.20_0a_f31.20_miscellaneous20_digital20_problems.20_0a_f32.20_conway27_27_s20_rats20_and20_palindromes.20_0a_index20_of20_authors20_cited20_0a_general20_index20_0a_glossary20_of20_symbols0a_0a_ k.="" b28.="" consecutive="" numbers="" with="" distinct="" prime="" factors.="" b29.="" is="" x="" determined="" by="" the="" divisors="" of="" _2b_="" _12c_="" _22c_.="" .="" _.2c_="" _k3f_="" b30.="" a="" small="" set="" whose="" product="" square.="" b31.="" binomial="" coeffcients.="" b32.="" _grimm27_27_s="" conjecture.="" b33.="" largest="" divisor="" coeffcient.="" b34.="" if="" _there27_27_s="" an="" i="" such="" that="" n="" -="" divides="" _nk_.="" b35.="" products="" same="" b36.="" _euler27_27_s="" totient="" function.="" b37.="" does="" _c3b6_28_n29_="" properly="" divide="" _13f_="" b38.="" solutions="" _c3b6_28_m29_="%c3%b3(n)." b39.="" _carmichael27_27_s="" b40.="" gaps="" between="" totatives.="" b41.="" iterations="" _c3b6_="" and="" _c3b3_.="" b42.="" behavior="" _c3b6_28_c3b3_28_n29_29_="" _c3b3_28_c3b6_28_n29_29_.="" b43.="" alternating="" sums="" factorials.="" b44.="" b45.="" euler="" numbers.="" b46.="" factor="" n.="" b47.="" when="" 2a="" -2b="" na="" _nb3f_="" b48.="" taken="" over="" primes.="" b49.="" smith="" c.="" additive="" number="" theory="" c1.="" _goldbach27_27_s="" c2.="" c3.="" lucky="" c4.="" ulam="" c5.="" determining="" members="" set.="" c6.="" addition="" chains.="" brauer="" hansen="" c7.="" money-changing="" problem.="" c8.="" sets="" subsets.="" c9.="" packing="" pairs.="" c10.="" modular="" di.erence="" error="" correcting="" codes.="" c11.="" three-subsets="" sums.="" c12.="" postage="" stamp="" c13.="" corresponding="" covering="" harmonious="" labelling="" graphs.="" c14.="" maximal="" sum-free="" sets.="" c15.="" zero-sum-free="" c16.="" nonaveraging="" nondividing="" c17.="" minimum="" overlap="" c18.="" queens="" c19.="" weakly="" indedendent="" sequence="" .nite="" union="" strongly="" independent="" _ones3f_="" c20.="" squares.="" c21.="" higher="" powers.="" d.="" diophantine="" equations="" d1.="" like="" d2.="" fermat="" d3.="" figurate="" d4.="" _waring27_27_s="" l="" kth="" d5.="" sum="" four="" cubes.="" d6.="" elementary="" solution="" x2="2y4" 1.="" d7.="" powers="" made="" power.="" d8.="" pyramidal="" equation.="" d9.="" catalan="" two="" d10.="" exponential="" equations.="" d11.="" egyptian="" fractions.="" d12.="" marko.="" d13.="" equation="" xxyy="zz." d14.="" ai="" bj="" d15.="" in="" pairs="" make="" d16.="" triples="" product.="" d17.="" blocks="" integers="" not="" d18.="" there="" perfect="" _cuboid3f_="" squares="" are="" differences="" d19.="" rational="" distances="" from="" corners="" d20.="" six="" general="" points="" at="" distances.="" d21.="" triangles="" integer="" _edges2c_="" medians="" area.="" d22.="" simplexes="" contents.="" d23.="" some="" quartic="" d24.="" equals="" d25.="" involving="" factorial="" d26.="" fibonacci="" various="" shapes.="" d27.="" congruent="" d28.="" reciprocal="" d29.="" m-tuples.="" e.="" sequences="" e1.="" thin="" all="" equal="" to="" member="" plus="" prime.="" e2.="" density="" l.c.m.="" each="" pair="" less="" than="" x.="" e3.="" comparable="" divisors.="" e4.="" no="" dividing="" r="" others.="" e5.="" divisible="" least="" one="" given="" e6.="" sequence.="" e7.="" series="" e8.="" e9.="" partitioning="" into="" classes="" numerous="" e10.="" theorem="" van="" der="" waerden.="" _szemerc2b4_edi27_27_s="" theorem.="" _classes3b_="" contains="" a.p.="" e11.="" _schur27_27_s="" classes.="" e12.="" version="" e13.="" e14.="" _rado27_27_s="" generalizations="" _waerden27_27_s="" problems.="" e15.="" recursion="" gobel.="" e16.="" 3x="" 1="" e17.="" permutation="" sequences.="" e18.="" _mahler27_27_s="" z-numbers.="" e19.="" parts="" fraction="" in.nitely="" often="" _prime3f_="" e20.="" davenport-schinzel="" e21.="" thue-morse="" e22.="" cycles="" containing="" permutations="" as="" subsequences.="" e23.="" a.p.s.="" e24.="" irrationality="" e25.="" _golomb27_27_s="" self-histogramming="" e26.="" _epstein27_27_s="" put-or-take-a-square="" game.="" e27.="" max="" mex="" e28.="" b2-sequences.="" mian-chowla="" e29.="" e30.="" _macmahon27_27_s="" measurement.="" e31.="" three="" hofstadter.="" e32.="" b2-sequences="" greedy="" algorithm.="" e33.="" monotone="" e34.="" happy="" e35.="" kimberling="" shuffle.="" e36.="" klarner-rado="" e37.="" mousetrap.="" e38.="" odd="" f.="" none="" above="" f1.="" _gauc39f_27_27_s="" lattice="" point="" f2.="" f3.="" _points2c_="" on="" circle.="" f4.="" no-three-in-line="" f5.="" quadratic="" residues.="" f6.="" patterns="" f7.="" cubic="" analog="" bhaskara="" f8.="" residues="" di.erences="" f9.="" primitive="" roots="" f10.="" two.="" f11.="" distribution="" f12.="" how="" its="" inverse="" opposite="" _parity3f_="" f13.="" systems="" congruences.="" f14.="" exact="" systems.="" f15.="" problem="" r.="" l.="" graham.="" f16.="" f17.="" associated="" _c3a6_-function.="" f18.="" size="" f19.="" partitions="" primes="" maximum="" f20.="" continued="" f21.="" partial="" quotients="" or="" f22.="" algebraic="" unbounded="" quotients.="" f23.="" 2="" 3.="" f24.="" decimal="" digital="" f25.="" persistence="" number.="" f26.="" expressing="" using="" just="" ones.="" f27.="" generalization="" farey="" series.="" f28.="" determinant="" value="" one.="" f29.="" _congruences2c_="" which="" always="" solvable.="" f30.="" polynomial="" values="" distinct.="" f31.="" miscellaneous="" f32.="" _conway27_27_s="" rats="" palindromes.="" index="" authors="" cited="" glossary="" symbols="">

Editorial Reviews

From the reviews of the third edition: "This is the third edition of Richard Guy's well-known problem book on number theory . . The earlier editions have served well in providing beginners as well as seasoned researchers in number theory with a good supply of problems. . many of the problems from earlier editions have been expanded with more up-to-date comments and remarks. . There is little doubt that a new generation of talented young mathematicians will make very good use of this book . ." (P. Shiu, The Mathematical Gazette, Vol. 89 (516), 2005)"The earlier editions of this book are among the most-opened books on the shelves of many practicing number theorists. The descriptions of state-of-the-art results on every topic and the extensive bibliographies in each section provide valuable ports of entry to the vast literature. A new and promising addition to this third edition is the inclusion of frequent references to entries in the Online encyclopedia of integer sequences at the end of each topic." (Greg Martin, Mathematical Reviews, Issue 2005 h)