1. Prove that (2m+1)^2 + (2n+1)^2 = (2p)^3 has no integer solutions.
2. Prove that, if (2m+1)^2 == 1 (mod 8) holds always.
3. If f(x + y) = f(x) + f(y) for all real numbers. then f(0) = 0.
4. From q3, prove that, f(n) = nf(1) for all natural numbers n.
5. Prove 4, for all rational numbers.
6. Prove 4, for all real numbers.
7. Prove that, for function in 3, f(x)/x is always a constant. What is this constant equal to.
8. If f(x + y) = f(x) + f(y) for all real numbers, prove that f(x) - f(y) = f (x - y).
9. If f(x + y) = f(x) + f(y) for all real numbers, prove that f is an odd function.
10. If f(x + y) = f(x)f(y) for all real numbers, prove that f(0) = 1 (Do not consider trivial definition of f).
11. From q10, prove that, f(n) = [f(1)]^n for all natural numbers n.
12. Prove 11, for all rational numbers.
13. Like f(x)/x in q7, what is constant for this class of functions?
14. Consider f(xy) = f(x) + f(y) for all reals. Prove that f(1) = 0.
15. Prove that f(m^n) = nf(m) for function class defined in 14. Where n and m are natural numbers.
16. Prove 15, for positive rational numbers m and n.
17. Like f(x)/x in q7, what is constant for this class of functions?
18. Consider f(xy) = f(x)f(y) for all reals. Prove that f(0) = 0. (Do not consider trivial definitions)
19. For 18, prove that f(1) = 1. (Do not consider trivial definitions)
20. If f(p) for a prime number p is always a composite number, prove that the f in 18, does not have any prime numbers as values.
21. If f is a family of linear functions, such that, any two functions f1 and f2 from f, satisfy the condition that, f1(f2(x)) == f2(f1(x)). Derive the condition that ensures this property, for family f.
22. What is such a condition for quadratic class of functions. (Quadratic polynomial necessarily will have non-zero leading coefficient).
23. Prove that if a1^(a2^x) = a2^(a1^x) for all x real, then a1 = a2.
24. log_a1(x) = log_x(a1), then what is x?
25. Prove that a^n + b^n = 2^n has no solutions apart from trivial ones.
26. Prove that a^2 - b^2 = c^3 has infinitely many solutions.
27. (2a+1)^2 + (2b+1)^2 = (2c)^3 has no solutions.
28. x^2 - y^2 = 6 has no solutions
29. x^m - y^m = 6 has no solutions
30. [UNSOLVED] Find x, y, m and n such that x^m - y^n = 6.
31. a^2 + b^2 = 2^n implies that a = b.
32. a^2 + b^2 = 2^n implies that a and b are even
33. Prove that a^2 - b^2 = c^n has infinitely many solutions.
34. Consider a function with this property.
f(x^m +/- y^n) = x^(m-1) * f(x) +/- y^(n-1) * f(y) (both m and n are at least 2).
Prove that f(0) = 0
35. Using the function specified in 34, prove that f(1) + f(-1) = 0
36. Using the function specified in 34, prove that f(2) = 2f(1)
37. Using the function specified in 34, prove that f(5) = 5f(1)
38. Using the function specified in 34, prove that f(3) = 3f(1)
39. Using the function specified in 34, prove that f(39) = 39f(1)
40. Prove that, for every odd number f is defined.
41. Prove that, for every even number of the form 4k, the function f is defined.
42. Prove that, all even numbers of the form 24k^2 + 2 have definition of f.
43. Prove that, all even numbers of the form (5k^2 + 375)/12 (of the form (4l+2) also) have definition of f.
44. Prove that, all even numbers of the form 18(k^2 + 3) have definition of f.
45. x^2 - y^2 = 14, no solution exists
46. x^m - y^m = 14, no solution exists
47. x^2 - y^2 = 30, no solution exists
48. x^m - y^m = 30, no solution exists
49. p^2 = 16k + 5, no solution exists
50. p^2 = 16k + 3, no solution exists
51. p^2 = 16k + 7, no solution exists
52. For which remainders (when p^2 is divided by 16) do solutions exist, and why?
53. When all squares are taken modulo 16, which moduli are more frequent than others?
54. Last digit in 2^2^2^.... 1000 times?
55. Consider a function with this property.
f(x^m +/- y^n +/- z^p) = x^(m-1) * f(x) +/- y^(n-1) * f(y) +/- z^(p-1) f(z) (m, n and p are at least 2).
Prove that f(0) = 0
56. Prove that f(6) = 6f(1).
57. Prove that f(14) = 14f(1).
*58. Is there a number n for which this function could not be defined?
59. Prove that if x^m - y^n = 6, then x and y are relatively prime.
60. Prove that if x^m - y^n = 6, then m and n are relatively prime.
61. Prove that if m^m - n^n = 6, has no solutions.
62. Prove that k^p - p^k = 6 has no solutions.
63. Prove that k^k - (k/2)^2k = 6 has no solutions.
64. Prove that k^2 = 0, 1 or 4 mod 8.
65. If f(x^n) = nx^(n-1)f(x), prove that f(1) = 0
66. For function in 65, prove also that f(0) = 0
67. For function in 65, prove also that f(4) = 2f(2)
68. For function in 65, prove also that f(x^2) = 2xf(x)
69. For function in 65, prove also that f(x^4) = 2x^2f(x^2)
70. For function in 65, prove also that f(x^mn) = mx^((m-1)*n)f(x^n)
71. Which function does the one in 65 remind you of?
72.
2. Prove that, if (2m+1)^2 == 1 (mod 8) holds always.
3. If f(x + y) = f(x) + f(y) for all real numbers. then f(0) = 0.
4. From q3, prove that, f(n) = nf(1) for all natural numbers n.
5. Prove 4, for all rational numbers.
6. Prove 4, for all real numbers.
7. Prove that, for function in 3, f(x)/x is always a constant. What is this constant equal to.
8. If f(x + y) = f(x) + f(y) for all real numbers, prove that f(x) - f(y) = f (x - y).
9. If f(x + y) = f(x) + f(y) for all real numbers, prove that f is an odd function.
10. If f(x + y) = f(x)f(y) for all real numbers, prove that f(0) = 1 (Do not consider trivial definition of f).
11. From q10, prove that, f(n) = [f(1)]^n for all natural numbers n.
12. Prove 11, for all rational numbers.
13. Like f(x)/x in q7, what is constant for this class of functions?
14. Consider f(xy) = f(x) + f(y) for all reals. Prove that f(1) = 0.
15. Prove that f(m^n) = nf(m) for function class defined in 14. Where n and m are natural numbers.
16. Prove 15, for positive rational numbers m and n.
17. Like f(x)/x in q7, what is constant for this class of functions?
18. Consider f(xy) = f(x)f(y) for all reals. Prove that f(0) = 0. (Do not consider trivial definitions)
19. For 18, prove that f(1) = 1. (Do not consider trivial definitions)
20. If f(p) for a prime number p is always a composite number, prove that the f in 18, does not have any prime numbers as values.
21. If f is a family of linear functions, such that, any two functions f1 and f2 from f, satisfy the condition that, f1(f2(x)) == f2(f1(x)). Derive the condition that ensures this property, for family f.
22. What is such a condition for quadratic class of functions. (Quadratic polynomial necessarily will have non-zero leading coefficient).
23. Prove that if a1^(a2^x) = a2^(a1^x) for all x real, then a1 = a2.
24. log_a1(x) = log_x(a1), then what is x?
25. Prove that a^n + b^n = 2^n has no solutions apart from trivial ones.
26. Prove that a^2 - b^2 = c^3 has infinitely many solutions.
27. (2a+1)^2 + (2b+1)^2 = (2c)^3 has no solutions.
28. x^2 - y^2 = 6 has no solutions
29. x^m - y^m = 6 has no solutions
30. [UNSOLVED] Find x, y, m and n such that x^m - y^n = 6.
31. a^2 + b^2 = 2^n implies that a = b.
32. a^2 + b^2 = 2^n implies that a and b are even
33. Prove that a^2 - b^2 = c^n has infinitely many solutions.
34. Consider a function with this property.
f(x^m +/- y^n) = x^(m-1) * f(x) +/- y^(n-1) * f(y) (both m and n are at least 2).
Prove that f(0) = 0
35. Using the function specified in 34, prove that f(1) + f(-1) = 0
36. Using the function specified in 34, prove that f(2) = 2f(1)
37. Using the function specified in 34, prove that f(5) = 5f(1)
38. Using the function specified in 34, prove that f(3) = 3f(1)
39. Using the function specified in 34, prove that f(39) = 39f(1)
40. Prove that, for every odd number f is defined.
41. Prove that, for every even number of the form 4k, the function f is defined.
42. Prove that, all even numbers of the form 24k^2 + 2 have definition of f.
43. Prove that, all even numbers of the form (5k^2 + 375)/12 (of the form (4l+2) also) have definition of f.
44. Prove that, all even numbers of the form 18(k^2 + 3) have definition of f.
45. x^2 - y^2 = 14, no solution exists
46. x^m - y^m = 14, no solution exists
47. x^2 - y^2 = 30, no solution exists
48. x^m - y^m = 30, no solution exists
49. p^2 = 16k + 5, no solution exists
50. p^2 = 16k + 3, no solution exists
51. p^2 = 16k + 7, no solution exists
52. For which remainders (when p^2 is divided by 16) do solutions exist, and why?
53. When all squares are taken modulo 16, which moduli are more frequent than others?
54. Last digit in 2^2^2^.... 1000 times?
55. Consider a function with this property.
f(x^m +/- y^n +/- z^p) = x^(m-1) * f(x) +/- y^(n-1) * f(y) +/- z^(p-1) f(z) (m, n and p are at least 2).
Prove that f(0) = 0
56. Prove that f(6) = 6f(1).
57. Prove that f(14) = 14f(1).
*58. Is there a number n for which this function could not be defined?
59. Prove that if x^m - y^n = 6, then x and y are relatively prime.
60. Prove that if x^m - y^n = 6, then m and n are relatively prime.
61. Prove that if m^m - n^n = 6, has no solutions.
62. Prove that k^p - p^k = 6 has no solutions.
63. Prove that k^k - (k/2)^2k = 6 has no solutions.
64. Prove that k^2 = 0, 1 or 4 mod 8.
65. If f(x^n) = nx^(n-1)f(x), prove that f(1) = 0
66. For function in 65, prove also that f(0) = 0
67. For function in 65, prove also that f(4) = 2f(2)
68. For function in 65, prove also that f(x^2) = 2xf(x)
69. For function in 65, prove also that f(x^4) = 2x^2f(x^2)
70. For function in 65, prove also that f(x^mn) = mx^((m-1)*n)f(x^n)
71. Which function does the one in 65 remind you of?
72.
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