если x и y - целые, тогда и (x-y-1) и (x+y+1) тоже целые.

Рассмотрим возможные разложения 91 на два целых множителя:

91 = 1·91 = 91·1 = (-1)·(-91) = (-91)·(-1)

91 = 7·13 = 13·7 = (-7)·(-13) = (-13)·(-7)

То есть у нас 8 возможностей, последовательно решаем их

1) x - y - 1 = 1 и x + y + 1 = 91

x - y = 2 и x + y = 90

2x = 92, x = 46, и y = 90 - x = 90 - 46 = 44.

(46; 44)

2) x - y - 1 = 91 и x+y+1 = 1

x-y = 92 и x+y = 0,

2x = 92, x = 46

y = -x = -46

(46; -46)

3) x - y - 1 = -1 и x+y+1 = -91

x-y = 0 и x+y = -92,

x = y

2x = -92,

x = -46,

y = -46

(-46; -46)

4) x - y - 1 = -91 и x+y+1 = -1

x - y = -90 и x+y = -2,

2x = -92

x = -46

y = -2-x = -2 +46 = 44.

(-46; 44)

5) x - y - 1 = 7 и x+y+1 = 13

x - y = 8 и x+y = 12

2x = 20, x = 10

y = 12 - x = 12 - 10 = 2

(10; 2)

6) x - y - 1 = 13 и x+y+1 = 7

x - y = 14 и x+y = 6,

2x = 20, x = 10,

y = 6 - x = 6 - 10 = -4,

(10; -4)

7) x - y - 1 = -7 и x+y+1 = -13,

x - y = -6 и x+y = -14,

2x = -20, x = -10

y = -14 - x = -14 + 10 = -4

(-10; -4)

8) x - y - 1 = -13 и x+y+1 = -7

x - y = -12 и x+y = -8,

2x = -20, x = -10,

y = -8-x = -8+10 = 2

(-10; 2).

Ответ. Мы нашли все решения: 8 пар целых чисел.