1000

n x 2 + 1 = y 2 nx^{2} + 1 = y^{2} n x 2 + 1 = y 2

y 2 − n x 2 = 1 y^{2} - nx^{2} = 1 y 2 − n x 2 = 1

n n n

( x , y ) (x, y) ( x , y )

( b 2 − n a 2 ) ( d 2 − n c 2 ) = ( b d + n a c ) 2 − n ( b c + a d ) 2 (b^{2} - na^{2})(d^{2} - nc^{2}) = (bd + nac)^{2} - n(bc + ad)^{2} ( b 2 − n a 2 ) ( d 2 − n c 2 ) = ( b d + n a c ) 2 − n ( b c + a d ) 2

( b 2 − n a 2 ) ( d 2 − n c 2 ) = ( b d − n a c ) 2 − n ( b c − a d ) 2 (b^{2} - na^{2})(d^{2} - nc^{2}) = (bd - nac)^{2} - n(bc - ad)^{2} ( b 2 − n a 2 ) ( d 2 − n c 2 ) = ( b d − n a c ) 2 − n ( b c − a d ) 2

b 2 − n a 2 = 1 b^{2} - na^{2} = 1 b 2 − n a 2 = 1 and d 2 − n c 2 = 1 d^{2} - nc^{2} = 1 d 2 − n c 2 = 1

( b d + n a c ) 2 − n ( b c + a d ) 2 = 1 (bd + nac)^{2} - n(bc + ad)^{2} = 1 ( b d + n a c ) 2 − n ( b c + a d ) 2 = 1

( b d − n a c ) 2 − n ( b c − a d ) 2 = 1 (bd - nac)^{2} - n(bc - ad)^{2} = 1 ( b d − n a c ) 2 − n ( b c − a d ) 2 = 1 .

( a , b ) (a, b) ( a , b )

( c , d ) (c, d) ( c , d )

( b c + a d , b d + n a c ) (bc + ad, bd + nac) ( b c + a d , b d + n a c ) and ( b c − a d , b d − n a c ) (bc - ad, bd - nac) ( b c − a d , b d − n a c ) .

( a , b (a, b ( a , b

)

( c , d ) (c, d) ( c , d )

n a 2 + k = b 2 na^{2} + k = b^{2} n a 2 + k = b 2 and n c 2 + k ′ = d 2 nc^{2} + k' = d^{2} n c 2 + k ′ = d 2

( b c + a d , b d + n a c ) (bc + ad, bd + nac) ( b c + a d , b d + n a c ) and ( b c − a d , b d − n a c ) (bc - ad, bd - nac) ( b c − a d , b d − n a c )

n x 2 + k k ′ = y 2 nx^{2} + kk' = y^{2} n x 2 + k k ′ = y 2 .

17

th

(

)

628

( a , b ) (a, b) ( a , b )

( 2 a b , b 2 + n a 2 ) (2ab, b^{2} + na^{2}) ( 2 a b , b 2 + n a 2 )

( a , b ) (a, b) ( a , b )

( a , b ) (a, b) ( a , b )

( a , b ) (a, b) ( a , b )

( 2 a b , b 2 + n a 2 ) (2ab, b^{2}+ na^{2}) ( 2 a b , b 2 + n a 2 )

x = a , y = b x = a, y = b x = a , y = b

n x 2 + k = y 2 nx^{2} + k = y^{2} n x 2 + k = y 2

( a , b ) (a, b) ( a , b )

( a , b ) (a, b) ( a , b )

( 2 a b , b 2 + n a 2 ) (2ab, b^{2} + na^{2}) ( 2 a b , b 2 + n a 2 )

n x 2 + k 2 = y 2 nx^{2} + k^{2} = y^{2} n x 2 + k 2 = y 2

k 2 k^{2} k 2

x = 2 a b k , y = b 2 + n a 2 k x = \large \frac {2ab}{k}

ormalsize, y = \large \frac {b^{2} + na^{2}}{k} x = k 2 a b ​ , y = k b 2 + n a 2 ​

n x 2 + 1 = y 2 nx^{2} + 1 = y^{2} n x 2 + 1 = y 2

x x x

y y y

k = 2 k = 2 k = 2

( a , b ) (a, b) ( a , b )

n x 2 + k = y 2 nx^{2} + k = y^{2} n x 2 + k = y 2

n a 2 = b 2 − 2 na^{2} = b^{2} - 2 n a 2 = b 2 − 2

x = 2 a b 2 = a b x = \large \frac{ 2ab}{2}

ormalsize = ab x = 2 2 a b ​ = a b ,



y = b 2 + n a 2 2 = 2 b 2 − 2 2 = b 2 − 1 y = \large \frac {b^{2} + na^{2}}{2}

ormalsize = \large \frac {2b^{2} - 2}{2}

ormalsize = b^{2} - 1 y = 2 b 2 + n a 2 ​ = 2 2 b 2 − 2 ​ = b 2 − 1

k = − 2 k = -2 k = − 2

k = 4 k = 4 k = 4

k = − 4 k = -4 k = − 4

( a , b ) (a, b) ( a , b )

n a 2 + k = b 2 na^{2} + k = b^{2} n a 2 + k = b 2

k k k

1

1

2

2

4

4

k k k

23 x 2 + 1 = y 2 23x^{2} + 1 = y^{2} 2 3 x 2 + 1 = y 2

a = 1 , b = 5 a = 1, b = 5 a = 1 , b = 5

23 a 2 + 2 = b 2 23a^{2} + 2 = b^{2} 2 3 a 2 + 2 = b 2

x = 5 , y = 24 x = 5, y = 24 x = 5 , y = 2 4

(5

24)

x = 2 × 5 × 24 = 240 , y = 2 4 2 + 23 × 5 2 = 1151 x = 2\times 5\times 24 = 240, y = 24^{2} + 23\times 5^{2} = 1151 x = 2 × 5 × 2 4 = 2 4 0 , y = 2 4 2 + 2 3 × 5 2 = 1 1 5 1

x = 11515 , y = 55224 x = 11515, y = 55224 x = 1 1 5 1 5 , y = 5 5 2 2 4

x = 552480 , y = 2649601 x = 552480, y = 2649601 x = 5 5 2 4 8 0 , y = 2 6 4 9 6 0 1

83 x 2 + 1 = y 2 83x^{2} + 1 = y^{2} 8 3 x 2 + 1 = y 2

(1

9)

83 × 1 2 - 2 = 9 2

x = 9 , y = 82 x = 9, y = 82 x = 9 , y = 8 2 .

( x , y ) (x,y) ( x , y )

(9 , 82) ,

(1476 , 13447) ,

(242055 , 2205226) ,

(39695544 , 361643617) ,

(6509827161 , 59307347962) ,

(1067571958860 , 9726043422151) ,

(175075291425879 , 1595011813884802)

628

1150

n x 2 + 1 = y 2 nx^{2} + 1 = y^{2} n x 2 + 1 = y 2

( a , b ) (a, b) ( a , b )

n a 2 + k = b 2 na^{2} + k = b^{2} n a 2 + k = b 2

a a a

b b b

k k k

a , k a, k a , k

m , ( 1 , m ) m, (1, m) m , ( 1 , m )

n . 1 2 + ( m 2 − n ) = m 2 n.1^{2} + (m^{2} - n) = m^{2} n . 1 2 + ( m 2 − n ) = m 2 .

( a , b ) (a, b) ( a , b )

( 1 , m ) (1, m) ( 1 , m )

n ( a m + b ) 2 + ( m 2 − n ) k = ( b m + n a ) 2 n(am + b)^{2} + (m^{2} - n)k = (bm + na)^{2} n ( a m + b ) 2 + ( m 2 − n ) k = ( b m + n a ) 2 .

k k k

x = a m + b k , y = b m + n a k x = \large \frac {am + b}{k}

ormalsize, y = \large \frac {bm + na}{k} x = k a m + b ​ , y = k b m + n a ​

n x 2 + m 2 − n k = y 2 nx^{2} + \large \frac {m^{2} - n}{k}

ormalsize = y^{2} n x 2 + k m 2 − n ​ = y 2 .

a , k a, k a , k

m m m

a m + b am + b a m + b

k k k

(

)

m m m

a m + b am + b a m + b

k k k

m 2 − n m^{2} - n m 2 − n

b m + n a bm + na b m + n a

k k k

m m m

x = a m + b k , y = b m + n a k x = \large \frac {am + b}{k}

ormalsize, y = \large \frac {bm + na}{k} x = k a m + b ​ , y = k b m + n a ​

n x 2 + m 2 − n k = y 2 nx^{2} + \large \frac {m^{2} - n}{k}

ormalsize = y^{2} n x 2 + k m 2 − n ​ = y 2

m 2 − n k \large \frac {m^{2} - n}{k} k m 2 − n ​

m m m

a m + b am + b a m + b

k k k

m 2 − n m^{2} - n m 2 − n

m 2 − n k \large \frac {m^{2} - n}{k} k m 2 − n ​

1

1

2

2

4

4

n x 2 + 1 = y 2 nx^{2} + 1 = y^{2} n x 2 + 1 = y 2

m 2 − n k \large \frac {m^{2} - n}{k} k m 2 − n ​

x = a m + b k , y = b m + n a k x = \large \frac {am + b}{k}

ormalsize, y = \large \frac {bm + na}{k} x = k a m + b ​ , y = k b m + n a ​

n x 2 + m 2 − n k = y 2 nx^{2} + \large \frac {m^{2} - n}{k}

ormalsize = y^{2} n x 2 + k m 2 − n ​ = y 2

n a 2 + k = b 2 na^{2} + k = b^{2} n a 2 + k = b 2

(

)

n x 2 + t = y 2 nx^{2} + t = y^{2} n x 2 + t = y 2

t t t

1

1

2

2

4

4

61 x 2 + 1 = y 2 61x^{2} + 1 = y^{2} 6 1 x 2 + 1 = y 2 .

m m m

( m + 8 ) / 3 (m + 8)/3 ( m + 8 ) / 3

m 2 − 61 m^{2} - 61 m 2 − 6 1

m = 7 m = 7 m = 7

x = 5 , y = 39 x = 5, y = 39 x = 5 , y = 3 9

n x 2 − 4 = y 2 nx^{2} - 4 = y^{2} n x 2 − 4 = y 2

x = 226153980 , y = 1766319049 x = 226153980, y = 1766319049 x = 2 2 6 1 5 3 9 8 0 , y = 1 7 6 6 3 1 9 0 4 9

61 x 2 + 1 = y 2 61x^{2} + 1 = y^{2} 6 1 x 2 + 1 = y 2

12

th

n x 2 + k = y 2 nx^{2} + k = y^{2} n x 2 + k = y 2

k k k

1

2

2

4

4

k k k

1

2

2

4

4

(

)

14

th

103 x 2 + 1 = y 2 103x^{2} + 1 = y^{2} 1 0 3 x 2 + 1 = y 2 .

a = 1 , b = 10 a = 1, b = 10 a = 1 , b = 1 0

103 × 1 2 − 3 = 1 0 2 103\times 1^{2} - 3 = 10^{2} 1 0 3 × 1 2 − 3 = 1 0 2 .

m m m

m + 10 m + 10 m + 1 0

3

m 2 − 103 m^{2} - 103 m 2 − 1 0 3

m = 11 m = 11 m = 1 1

103 × 7 2 − 6 = 7 1 2 103\times 7^{2} - 6 = 71^{2} 1 0 3 × 7 2 − 6 = 7 1 2 .

m m m

7 m + 71 7m + 71 7 m + 7 1

6

m 2 − 103 m^{2} - 103 m 2 − 1 0 3

m = 7 m = 7 m = 7

103 × 2 0 2 + 9 = 20 3 2 103\times 20^{2} + 9 = 203^{2} 1 0 3 × 2 0 2 + 9 = 2 0 3 2 .

m m m

20 m + 203 20m+203 2 0 m + 2 0 3

9

m 2 − 103 m^{2} - 103 m 2 − 1 0 3

m = 11 m = 11 m = 1 1

103 × 4 7 2 + 2 = 47 7 2 103\times 47^{2} + 2 = 477^{2} 1 0 3 × 4 7 2 + 2 = 4 7 7 2 .

k = 2 k = 2 k = 2

x = 22419 , y = 227528 x = 22419, y = 227528 x = 2 2 4 1 9 , y = 2 2 7 5 2 8 .

97 x 2 + 1 = y 2 97x^{2} + 1 = y^{2} 9 7 x 2 + 1 = y 2

97 × 1 2 + 3 = 1 0 2 97\times 1^{2} + 3 = 10^{2} 9 7 × 1 2 + 3 = 1 0 2



97 × 7 2 + 8 = 6 9 2 97\times 7^{2} + 8 = 69^{2} 9 7 × 7 2 + 8 = 6 9 2



97 × 2 0 2 + 9 = 19 7 2 97\times 20^{2} + 9 = 197^{2} 9 7 × 2 0 2 + 9 = 1 9 7 2



97 × 5 3 2 + 11 = 52 2 2 97\times 53^{2} + 11 = 522^{2} 9 7 × 5 3 2 + 1 1 = 5 2 2 2



97 × 8 6 2 − 3 = 84 7 2 97\times 86^{2} - 3 = 847^{2} 9 7 × 8 6 2 − 3 = 8 4 7 2



97 × 56 9 2 − 1 = 560 4 2 97\times 569^{2} - 1 = 5604^{2} 9 7 × 5 6 9 2 − 1 = 5 6 0 4 2

x = 6377352 , y = 62809633 x = 6377352, y = 62809633 x = 6 3 7 7 3 5 2 , y = 6 2 8 0 9 6 3 3

17

th

1657

We await these solutions, which, if England or Belgic or Celtic Gaul do not produce, then Narbonese Gaul will.

500

61 x 2 + 1 = y 2 61x^{2} + 1 = y^{2} 6 1 x 2 + 1 = y 2 .

1657

58

1658

n n n

150

313 x 2 + 1 = y 2 313x^{2} + 1 = y^{2} 3 1 3 x 2 + 1 = y 2 .

x = 1819380158564160 , y = 32188120829134849 x = 1819380158564160, y = 32188120829134849 x = 1 8 1 9 3 8 0 1 5 8 5 6 4 1 6 0 , y = 3 2 1 8 8 1 2 0 8 2 9 1 3 4 8 4 9

( b 2 − n a 2 ) ( d 2 − n c 2 ) = ( b d + n a c ) 2 − n ( b c + a d ) 2 (b^{2} - na^{2})(d^{2} - nc^{2}) = (bd + nac)^{2} - n(bc + ad)^{2} ( b 2 − n a 2 ) ( d 2 − n c 2 ) = ( b d + n a c ) 2 − n ( b c + a d ) 2 .

1658

1685

98

1658

n x 2 + 1 = y 2 nx^{2} + 1 = y^{2} n x 2 + 1 = y 2

n n n

n n n

1766

1771

n n n

√ n √n √ n

19

19 = 4 + 1 2 + 1 1 + 1 3 + 1 1 + 1 2 + 1 8 + 1 2 + 1 1 + 1 3 + 1 1 + 1 2 + 1 8 + 1 2 + 1 1 + . . . \sqrt{19} = 4+ \frac{1}{2+} \frac{1}{1+} \frac{1}{3+} \frac{1}{1+} \frac{1}{2+} \frac{1}{8+} \frac{1}{2+} \frac{1}{1+} \frac{1}{3+} \frac{1}{1+} \frac{1}{2+} \frac{1}{8+} \frac{1}{2+} \frac{1}{1+} ... 1 9 ​ = 4 + 2 + 1 ​ 1 + 1 ​ 3 + 1 ​ 1 + 1 ​ 2 + 1 ​ 8 + 1 ​ 2 + 1 ​ 1 + 1 ​ 3 + 1 ​ 1 + 1 ​ 2 + 1 ​ 8 + 1 ​ 2 + 1 ​ 1 + 1 ​ . . .

19 = 4 + 1 1 + 1 2 + 1 1 + 1 3 + 1 2 + 1 8 + 1 2 + 1 1 + . . . \sqrt{19} = 4+ \large\frac{1}{1+ \large\frac{1}{2+\large\frac{1}{1 + \large\frac{1}{3 + \large\frac{1}{2 + \large\frac{1}{8 + \large\frac{1}{2 + \large\frac{1}{1 + ...}}}}}}}} 1 9 ​ = 4 + 1 + 2 + 1 + 3 + 2 + 8 + 2 + 1 + . . . 1 ​ 1 ​ 1 ​ 1 ​ 1 ​ 1 ​ 1 ​ 1 ​

6

170 39 \large\frac{170}{39}

ormalsize 3 9 1 7 0 ​

x = 39 , y = 170 x = 39, y = 170 x = 3 9 , y = 1 7 0

19 x 2 + 1 = y 2 19x^{2} + 1 = y^{2} 1 9 x 2 + 1 = y 2 .

170

39

19

( 170 + 39 √ 19 ) 2 = 57799 + 13260 √ 19 (170 + 39√19)^{2} = 57799 + 13260√19 ( 1 7 0 + 3 9 √ 1 9 ) 2 = 5 7 7 9 9 + 1 3 2 6 0 √ 1 9

x = 13260 , y = 57799 x = 13260, y = 57799 x = 1 3 2 6 0 , y = 5 7 7 9 9

( 170 + 39 √ 19 ) 3 = 19651490 + 4508361 √ 19 (170 + 39√19)^{3} = 19651490 + 4508361√19 ( 1 7 0 + 3 9 √ 1 9 ) 3 = 1 9 6 5 1 4 9 0 + 4 5 0 8 3 6 1 √ 1 9

x = 4508361 , y = 19651490 x = 4508361, y = 19651490 x = 4 5 0 8 3 6 1 , y = 1 9 6 5 1 4 9 0

(170

39

19)

19

2

1

2

57799 + 13260 √ 19



19651490 + 4508361 √ 19



6681448801 + 1532829480 √ 19



2271672940850 + 521157514839 √ 19



772362118440199 + 177192022215780 √ 19



262600848596726810 + 60244766395850361 √ 19



89283516160768675201 + 20483043382566906960 √ 19

n n n

n n n

n n n

n n n

(

)