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Back Matter

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Classical Unsolved Problems

  1. Is every even number > 2 the sum of two primes? (Goldbach)
  2. Is every number of the form 4n+2 (n>1) the sum of two primes of the form 4n+1? (Euler)
  3. Obtain an asymptotic formula for the number of representations of 2n as the sum of two primes.
  4. Can every even number be expressed as the difference of two primes?
  5. Can every even number be expressed as the difference of two primes in infinitely many ways?
  6. In particular, are there infinitely many prime pairs?
  7. Find an asymptotic formula for the number of prime pairs x.
  8. Do there exist infinitely many primes of the form x2+1?
  9. Does any polynomial of degree > 1 represent infinitely many primes?
  10. Are there infinitely many Fermat primes?
  11. Are there infinitely many Mersenne primes (primes of the form 2p1)?
  12. Are there infinitely many primes of the form 2p+1, where p is a prime?
  13. Is there at least one prime between every pair of consecutive squares?
  14. Are there odd perfect numbers?
  15. Are there infinitely many multiply perfect numbers?
  16. Are there infinitely many pairs of amicable numbers?
  17. Let f(n)=σ(n)n. Does the sequence f0(n)=n, fk+1(n)=f(fk(n)), k=1,2,3,... remain bounded for every n? (Poulet)
  18. Are there infinitely many primes p for which 2p11 is divisible by p2?
  19. Are there infinitely many primes p for which (p1)!+1 is divisible by p2?
  20. Is xn+yn=zn solvable for every n>2? (Fermat)
  21. Is xn1+xn2++xnn1=xnn solvable for any n>2? (Euler)
  22. Is 2 a primitive root of infinitely many primes? (Artin conjectures that 2 is a primitive root of about one third of all primes.)
  23. is Euler's constant γ irrational?
  24. Is πe irrational?
  25. Are 8 and 9 the only powers (exceeding 1) of integers that differ by 1? (Catalan.)
  26. For what values of k is x2+k=y3?

Miscellaneous Problems

  1. Show that nd=1(2d1)nd=nd=1nd2.
  2. Show that d | nτ(d)3=(d | nτ(d))2.
  3. Show that (a,b)=11(ab)2=52.
  4. Show that p2+1p21=52.(The product runs over all primes.)
  5. Generalize the results of Problems 3 and 4 above.
  6. Show that limnd | (n!+1)1d=1.
  7. Show that limnd | Fn1d=1, where Fn=22n+1.
  8. Prove that π(x)=xn=1nj=1e2πi((n1)!+1)j/n.
  9. Prove that (a,b)=a1m=0a1n=11ae2πibmn/a.
  10. Show that the least absolute remainder of a (mod b) is ab2ab+bab.
  11. Prove that n=1φ(n)n! is irrational.
  12. Prove that n=1σ(n)n! is irrational.
  13. Prove that n=1σ2(n)n! is irrational.
  14. Show that xn=1nσ(n)x+1.
  15. Show that d2 | nμ(d)=|μ(n)|.
  16. Show that for g3,1+g+g2+g3+g4 is not a square.
  17. For n an integer and a0 prove that n1k=1a+kn=na.
  18. Show that 1ϕ(n)=1nd | nμ2(d)ϕ(d).
  19. Prove that n=1μ(n)xn1+xn=x2x2.
  20. Prove that λ(n)xn1xn=n=1xn2.
  21. Prove that F(n)=d | nf(d) if and only if f(n)=d | nF(nd)μ(d).
  22. Show that the sum of the odd divisors of n is d | n(1)(nd)d.
  23. Prove that the product of the integers n and relatively prime to n is nφ(n)d | n(d!dd)μ(nd).
  24. Show that every integer has a multiple of the form 11 ... 1100 ... 00.
  25. Prove that there are infinitely many square-free numbers of the form n2+1.
  26. Prove that \({m \choose 0} + {m \choose 3} + {m \choose 6} + \cdot\cdot\cdot \not\equiv 0) (mod 3).
  27. Show that the number of representations of n as the sum of one or more consecutive positive integers is τ(n1) where n1 is the largest odd divisor of n.
  28. Prove that φ(x)=n! is solvable for every n.
  29. Prove that φ(x)=27n is not solvable for any positive n.
  30. Prove that 30 is the largest integer such that every integer less than it and relatively prime to it is 1 or a prime.
  31. Let a,b,x be integers and let x0=x, xn+1=axn+b n>0. Prove that not all the x’s are primes.
  32. Show that the only solutions of φ(n)=τ(n) are n=2,3,4,8,14,20,90.
  33. Show that φ(n+1)=pn+1pn is valid only for 1n5.
  34. Show that (2a)!(2b)!a!b!(a+b)! is an integer.
  35. show that if (a,b)=1 then (a+b1)!a!b! is an integer.
  36. Show that an integral polynomial of at least the first degree cannot represent only primes.
  37. Show that if f(x) is an integral polynomial of degree > 0, then f(x) for x=1,2,... has an infinite number of distinct prime divisors.
  38. Find the number of integers prime to m in the set 12,23,...,m(m+1).
  39. Prove that the Fermat numbers are relatively prime in pairs.
  40. Let T1=2, Tn+1=T2nTn1. Prove that Ti,Tj)=1,ij.
  41. Prove that 2ζ(3)=n=11n2(1+12+13++1n).
  42. Prove that the density of numbers for which (n,φ(n))=1 is zero.
  43. Show that for some n, 2n has 1000 consecutive 7’s in its digital representation.
  44. Prove that infinitely many squares do not contain the digit 0.
  45. Show that for some n, pn contains 1000 consecutive 7’s in its digital representation.
  46. Show that the density of the numbers n for which φ(x)=n is solvable is zero.
  47. Show that if φ(x)=n has exactly one solution then n>10100.
  48. Prove that ep(n)=nsp(n)p1.
  49. Let a1, a2, ..., ap1 be ordered by not necessarily distinct nonzero residue classes (mod p). Prove that there exist 1ijp1 such that aiai+1aj1 (mod p).
  50. Show that the nth prime is the limit of the sequence.
    n0=n,nk+1=n0+π(n0+n1++nk).
  51. Show that the nth nonprime is the limit of the sequence
    n,n+π(n),n+π(n+π(n)),....
  52. Prove that every positive integer is either of the form n+π(n1) or of the form n+pn, but not both.
  53. Show that (3+22)2n1+(322)2n12 is square for every n1.
  54. Prove that for every real ϵ>0 there exists a real α such that the fractional part of αn is greater than 1ϵ for every integer n>0.
  55. Show that if p and q are integers n, then it is possible to arrange n or fewer unit resistances to give a combined resistance of pq.
  56. Show that (a,n)=1 and x=a12k1k[kan] imply ax1 (mod n).
  57. If (a,b)=d prove that a1x=1[bxa]=(a1)(b1)2+d12.
  58. Show that the sum of reciprocals of integers representable as sums of two squares is divergent.
  59. Show that the sum of reciprocals of integers whose digital representation does not include 1000 consecutive 7’s is convergent.
  60. Prove that every n>1 can be expressed as the sum of two deficient numbers.
  61. Prove that every n>105 can be expressed as the sum of two abundant numbers.
  62. Prove that every sufficiently large n can be expressed as the sum of two k-abundant numbers.
  63. Prove that the nth nonsquare is n+n. {x} denotes the integer closest to x.)
  64. Prove that the nth nontriangular number is n+2n.
  65. Prove that the nth non-kth power is
    n+kn+kn.
  66. Show that the binary operation defined on nonnegative integers by
    mn=m+n+2mn
    is associative.
  67. Prove the same for the operation m×n=m+n+2mn.
  68. Prove that for p>5, (p1)!+1 contains a prime factor p.
  69. Show that the only solutions of (n1)!=nk1 are (n,k)=(2,1),(3,1) and (5, 2).
  70. Show that x2α22α1 (mod p) has a solution for every prime p if α3.
  71. Show that if f(x) is a polynomial with integer coefficients and f(a) is a square for each a, then f(x)=(g(x))2, where g(x) is a polynomial with integer coefficients.
  72. Given integers a1<a2<akn with kn2, prove that for some ijk, ai | aj.
  73. Show that two of the ai’s of Problem 72 are relatively prime.
  74. With the a’s of Problem 72, show that ai+aj=ak is solvable.
  75. Show that the number of solutions of x+2y+3z=n in non negative integers is
    (n+3)212.
  76. Show that the number of solutions of x+2y+4z=n in non negative integers is (n+2)(n+5)16+(1)nn16.
  77. Show that n and n + 2 are simultaneously prime if and only if
    m1n+2m+nmn+1mn1m=4.
  78. Show that n and n+2 are simultaneously prime if and only if
    4(n1)!+1+n0 (mod n(n+2)), (n>1).
  79. Show that for every n, 610n+2, and 1125102n+1±8 are Pythagorean triples.
  80. Show that the number of ordered pairs of integers whose l.c.m. is n is τ(n2).
  81. Show that 12+13++1n is never an integer.
  82. Show that x2+2y22x2+y2 is a square if and only if x=y.
  83. Prove that
    n=1φ(n)xn1+xn=x(1+x2)(1x2)2.
  84. show that the number of regular n-gons of unit edge is φ(n)2.
  85. Prove that the nth order determinant with aij=(i,j) has value ni=1φ(i).
  86. Prove that
    ni=1i=n3n+1n23.
  87. Prove that if p=4n+3 and q=8n+7 are both prime then q | 2p1.
  88. Show how to split the positive integers into two classes so that neither class contains all the positive terms of any arithmetic progression with common difference exceeding 1.
  89. Show that the reciprocal of every integer n>1 can be expressed as the sum of a finite number of consecutive terms of the form 1j(j+1)
  90. In how many ways can this be done? (Answer: 12(τ(n2)1).)
  91. Show that every rational can be expressed as a sum of a finite number of distinct reciprocals of integers.
  92. Show that the density of integers for which (n, n) = 1 is 1π2.
  93. Show that the expected value of (n, n) is π26.
  94. Prove that x2a (mod p) for every prime p implies that a is a square.
  95. Prove that f(a,b)=f(a)f(b) for (a,b)=1 and f(a+1)f(a) for every a imply that f(a)=ak.
  96. Find all primes in the sequence 101, 10101, 1010101, . . . .
  97. Find all primes in the sequence 1001, 1001001, 1001001001, . . . .
  98. Show that if f(x)>0 for all x and f(x)0 as x then there exists at most a finite number of solutions in integers of f(m)+f(n)+f(p)=1.
  99. Prove that the least nonresidue of every prime p>23 is less than p.
  100. Prove the existence of infinite sequences of 1’s, 2’s, and 3’s, no finite part of which is immediately repeated.
  101. Let d(n) denote the number of square divisors of n. Prove that
    limn1nnm=1d(m)=π26.
  102. Find all r such that n! cannot end in r zeros.
  103. Let a1,a2,...,an be integers with a1=1 and ai<ai+12ai. Prove that there exists a sequence ϵi of ±1's such that ni=1ϵiai=0 or 1.
  104. Show that for p a prime, p1 (mod 4)
    p+2p++p14p=p2112.
  105. Prove that π2 is irrational.
  106. Prove that cospq is irrational.
  107. If nin1n2...ni1 prove that 1ni is irrational.
  108. Prove that ae2+be+c0 if a,b,c are integers.
  109. Prove that
    τ(n)=nn1+2n1d=1(ndn1d).
  110. Let n=a0+a1p+a2p2++akpk where p is a prime and 0ai<p. Show that the number of binomial coefficients of order n that are relatively prime to p is (ai+1).
  111. Show that if r1,r2,...,rp1 form a complete residue system (mod p) then 1r1,2r2,...,(p1)rp1 do not.
  112. Show that 3 is a primitive root of every Fermat prime.
  113. Show that the number of ways in which n can be represented as the product of two relatively prime factors is 2ω(n)1.
  114. Prove that every even perfect number is of the form 2p1(2p1).
  115. Show that if f(x) is a polynomial with integral coefficients and there are ψ(m) integers relatively prime to m in the set f(1),f(2),...,f(m) then ψ is a weakly multiplicative function.
  116. If p=4n+1 is a prime, show that (2n)!2+10 (mod p).
  117. Show that 128 is the largest integer not representable as the sum of distinct squares.
  118. Show that x3+y4=z5 has infinitely many solutions.
  119. Show that xn+yn=xn+1 has infinitely many solutions.
  120. Show that for every k>0 there exists a lattice point (x1,y1) such that for every lattice point (x,y) whose distance from (x1,y1) does not exceed k, the g.c.d. (x,y)>1.
  121. Prove that no four distinct squares are in arithmetic progression.
  122. Prove that for n composite, π(n)<nlogn.
  123. Prove that 2n | (n+5)n.
  124. Prove that an odd p is prime if and only if p+k2 is not a square for k=1,2,...,p32.

Unsolved Problems and Conjectures

  1. Does φ(n)=φ(n+1) have infinitely many solutions?
  2. Does σ(n)=σ(n+1) have infinitely many solutions?
  3. Does φ(n)=φ(n+1)==φ(n+k) have solutions for every k? (Erd¨os)
  4. Conjecture: There is no n for which φ(x)=n has a unique solution. (Carmichael)
  5. Conjecture: For every positive integer k>1 there exist infinitely many n for which φ(x)=n has exactly k solutions.
  6. Do there exist solutions of σ(n)=2n+1?
  7. Is φ(x)=φ(y)=2n solvable for every n? (Moser)
  8. Are there infinitely many solution of τ(n)=τ(n+1)?
  9. Are there infinitely many numbers not of the form ϕ(n)+n? (Erd¨os)
  10. Are there infinitely many numbers not of the form σ(n)+n? (Erd¨os)
  11. Do there exist solutions of σ(x)=mσ(y) for every integer m? (Sierpinski)
  12. Are 1, 2, 4, 8, and 128 the only powers of 2, all of whose digits are powers of 2? (Starke)
  13. Does there exist for every n, n distinct integers all of whose sums in pairs are squares? (This is true for n5.)
  14. Does there exist a sequence of ϵi of ±1's such that ni=1ϵik is bounded for every k? (Erd¨os)
  15. If f(n) is an arithmetic function of period k and not identically 0, is it true that f(n)n0? (Erd¨os)
  16. Conjecture: For n sufficiently large, n can be partitioned n=a+b+c+d=d+e+f with abc=def. (Motzkin)
  17. Is n=1σk(n)n! irrational for every k? (Erd¨os and Kac)
  18. Is 1x+1y+1z=4n sovable for every n? (Erd¨os)
  19. Has n!+1=x2 any solutions with n>7? (Brochard)
  20. Is (2n)!(n+2)!2 an integer for infinitely many n? (Erd¨os)
  21. Is (2n)!n!(n+k)! an integer for every k and infinitely many n? (Erd¨os)
  22. Does there exist an A such that An is prime for every n? (Mills)
  23. Does en represent infinitely many primes?
  24. Does en represent infinitely many composite numbers? (Erd¨os)
  25. The number 105 has the property that 1052n is prime whenever it is positive. Is 105 the largest number with this property?
  26. Is 968 the largest number n such that for all k with (n,k)=1 and n>k2, nk2 is prime? (Erd¨os)
  27. Does there exist a prime p>41 such that x2x+p is prime for 1xp1?
  28. Let α(n) denote the number of 1’s in the binary representation of n. Does there exist a k such that for infinitely many primes p, α(p)<k? (Bellman)
  29. If f(x) is a polynomial with integer coefficients, f0(a)=a, and fn+1(a)=f(fn(a)), can a sequence fn(a), n=1,2,... consist entirely of primes?
  30. For p sufficiently large and ab0, n>2, does the polynomial xn+ax+b assume more than p2 values (mod p)? (Chowla)
  31. Find pairs of integers m, n such that m, n have the same prime factors and m+1, n+1 have the same prime factors; e.g., m=2k2 and n=2k(2k2). Are these the only cases? (Strauss)
  32. What is the largest integer not representable as the sum of distinct cubes?
  33. Let 1<u1<u2< be the sequence of integers of the form x2+y2. Conjecture:
    limnun+1unu1/4n=0. (Chowla and Davenport)
  34. Conjecture: |nx(1)n1pn|px2. (Pillai)
  35. Can every prime p3 (mod 8), p>163, be written as the sum of three distinct squares? (Chowla)
  36. Is ζ(3) irrational? Is ζ(2s+1) irrational?
  37. Conjecture: The only solution of 1n+2n++mn=(m+1)n is 1 + 2 = 3. (Bowen)
  38. Conjecture: the only solutions of an+(a+1)n++(a+b)n=(a+b+1)n are 1+2=3, 32+42=52, and 33+43+53=63. (Erd¨os)
  39. Does the equation 12+22++m2=(m+1)2++n2 have solutions? (Kelly)?
  40. The product of n>1 consecutive integers is not a kth power.
  41. Conjecture: If α>0 is not an integer then the density of solutions of (n,nα = 1\) is 6/π2. (Lambek and Moser)
  42. Conjecture: The only solutions of
    1x11x2++1xn+1x1x2...xn=1
    are3
    12+12=12+13+16=12+13+17+142=1 (Erd¨os)
  43. Is it true that for all pairs of primes p, q all sufficiently large numbers can be written as the sum of distinct numbers of the form pαqβ? (Erd¨os)
  44. Let a1,a2,... be integers not exceeding n such that the l.c.m. of any two is >n. What in the maximum of 1ai? Conjecture: 31/30. (Erd¨os)
  45. Let 0<a1<a2<<akn be such that the sums of distinct ai's are distinct. conjecture: klog2n is bounded. (Erd¨os)
  46. Give a relatively simple proof of Van der Waerden’s theorem for the case of two classes.
  47. Give a relatively simple proof of Roth’s theorem: Any sequence that does not contain an arithmetic progression has zero density.
  48. Give an elementary proof of Dirichlet’s theorem on quadratic residues:
    (np)>0 for p3 (mod 4).
  49. Let a1<a2<... be a sequence of positive integers and let f(n) denote the number of solutions of ai+aj=n. Conjecture: If f(n)>0 for every n then f(n) is unbounded. (Erd¨os and Turan)
  50. If the f(n) of Problem 49 is > 0 for every n then every sufficiently large n can be written as the sum of three distinct a’s. (Kelly)
  51. Construct a sequence of a’s for which the f(n) of Problem 49 is > 0 and for which f(n)<logn for every n. (Erd¨os has shown that such sequences exist.)
  52. Does there exist a sequence A with counting function A(n)<cn/logn such that every integer can be represented in the form a+2i,aA?
  53. Improve the bound [n!e] in Schur’s theorem in combinatorial number theory.
  54. Conjecture. If a1<a2< is a sequence of integers with an/an+11 and if for every d, every residue (mod d) is representable as the sum of distinct a’s, then at most a finite number of integers are not representable as the sum of distinct a’s. (Erd¨os)
  55. Is the sum of the reciprocals of those integers that are representable as the sum of k kth powers divergent? (Klamkin and Newman)
  56. Conjecture: For every ϵ>0 there exits an N=N(ϵ) such that for n>N the n-dimensional game of tic-tac-toe played on a 3×3××3 board must terminate before ϵ3n moves have been played. (Moser)
  57. Same as Problem 56 with 3 replaced by k.
  58. Every integer belongs to one of the arithmetic progressions {2n}, {3n}, {4n+1}, {6n+5}, {12n+7}, n=1,2,.... This is the simplest example of a finite set of arithmetic progressions, each with different common difference, all of whose common differences are greater than one, that contain all integers. Does there exist for every c>0 such a set of progressions, each common difference being >c? (Erd¨os)
  59. Give an explicit representation of n as the sum of four squares.
  60. Do there exist for every n, n primes that are consecutive terms of an arithmetic progression?
  61. Let 11+x+2x2=n=1anxn. Conjecture: |an|>cloglogn.
  62. Are there infinitely primes of the form 1111?
  63. Are there infinitely many Euclid primes 23pn+1?
  64. Conjecture: The least nonresidue of a prime p is <clogp.
  65. Conjecture: The least primitive root of a prime p is <pϵ, p>p0(ϵ).
  66. Conjecture: The number of perfect numbers n is <clogn.
  67. Find good bounds for the density of the abundant numbers.
  68. Prove that the ratio of residues to nonresidues in the range (1,[p]) approaches 1 as p.
  69. Give an elementary proof of pnp<3n.
  70. Conjecture: limn(an+1an)= implies n=1an2an irrational. (Erd¨os)
  71. Find all solutions of x4+y4=z4+t4.
  72. Find all solutions of x4+y4+z4=t4.
  73. Find all solutions of xxyy=zz.
  74. Let (n) be the least r for which there exists a chain of integers
    a0=1<a1<a2<<ar=n,
    where for each i>0, ai=aj+ak for some j, k<i (j=k permitted). Conjecture: (2q1)q+(q)1. (Scholz)
  75. Conjecture: (n)<(2n) for all n>0. (Utz)
  76. Let S(n) denote the number of solutions of (x)=n. Is it true that S(n)<S(n+1) for all n>0? (Utz)
  77. Polya's conjecture: xn=1λ(n)0, x>1. (Checked for x<800000.)
  78. Turan's conjecture: xn=1λ(n)n>0. (Checked for x<50000.)
  79. Pillai's conjecture: |xmyn|<N, m,n>1 has for every N only a finite number of solutions.
  80. 2e is irrational.
  81. Find a reasonable estimate for the number of solutions in positive integers of
    1x1+1x2+1x3++1xn=1.
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