Ο καθηγητής των Μαθηματικών λέει στον Πέτρο, το γινόμενο δύο ακεραίων αριθμών και στον Παύλο το άθροισμα τους. Κανείς τους δεν γνωρίζει τους αριθμούς που είπε ο καθηγητής στον άλλον.
Ένας από αυτούς λέει: Δεν μπορείς να μαντέψεις τον αριθμό που είπε σε μένα. Τότε, ο άλλος απαντά: Κάνεις λάθος, ο αριθμός είναι 136.
Ποιους αριθμούς είπε ο καθηγητής στον καθένα;
Η Πέτρος δεν μπορεί να είναι αυτός που κάνει την πρώτη δήλωση, μιας και θα μιλούσε μόνο αν ο αριθμός του ήταν πρώτος. Κι αυτό γιατί τότε θα μπορούσε να συμπεράνει πως ο Παύλος είχε τον αριθμό \(s=p+1\) (αφού ένας πρώτος αριθμός αναλύεται μόνο στο γινόμενο 1 επί τον εαυτό του). Άρα πρώτος μίλησε ο Παύλος.
H Παύλος τώρα, σκεπτόμενος ανάποδα, διαπίστωσε ότι αφαιρώντας \(1\) από τον αριθμό του δεν προκύπτει πρώτος αριθμός. Άρα ο Πέτρος δεν θα μπορούσε - με βάση το προηγούμενο σκεπτικό - να καταλάβει τον αριθμό του. Έτσι δήλωσε “Δεν μπορείς να μαντέψεις τον αριθμό που είπε σε μένα”.
Η Πέτρος όμως, εξαιτίας αυτής της δήλωσης του Παύλου, αντιλαμβάνεται ότι το \(s-1\), δηλαδή το \(x+y-1\), δεν είναι πρώτος.
Για να καταλήξει στη 2η δήλωση, σημαίνει ότι ο αριθμός που έχει μπορεί να αναλυθεί σε γινόμενο δύο αριθμών \(x\) και \(y\) με ένα και μόνο τρόπο ώστε να ισχύει ότι \(x+y-1\) δεν είναι πρώτος.
library(numbers)
library(spuRs)## Loading required package: MASS
## Loading required package: latticefor (x in 1:68) {
fact = c(1, primeFactors(x * (136 - x)))
print(fact)
ve <- vector()
for (i in 1:(length(fact) - 1)) {
total = fact
first <- combn(total, i, MARGIN=2)
second <- apply(first, 2, function(z) total[is.na(pmatch(total, z))])
if (is.null(dim(second)))
second <- matrix(second, ncol = length(second))
for (j in 1:ncol(first)) {
u = 0
u = u + 1 - prime(prod(first[, j]) + prod(second[, j]) - 1)
if (u == 1 & !(prod(first[, j]) %in% ve)) {
cat('x=',x," y=",136-x," first=",prod(first[, j])," second=",prod(second[, j]),"\n")}
ve <- c(ve,prod(first[, j]),prod(second[, j]))
}
}
}## [1] 1 3 3 3 5
## x= 1 y= 135 first= 1 second= 135
## [1] 1 2 2 67
## x= 2 y= 134 first= 1 second= 268
## x= 2 y= 134 first= 2 second= 134
## x= 2 y= 134 first= 67 second= 4
## [1] 1 3 7 19
## x= 3 y= 133 first= 1 second= 399
## x= 3 y= 133 first= 3 second= 133
## x= 3 y= 133 first= 7 second= 57
## x= 3 y= 133 first= 19 second= 21
## [1] 1 2 2 2 2 3 11
## x= 4 y= 132 first= 1 second= 528
## x= 4 y= 132 first= 2 second= 264
## x= 4 y= 132 first= 3 second= 176
## x= 4 y= 132 first= 11 second= 48
## x= 4 y= 132 first= 4 second= 132
## x= 4 y= 132 first= 6 second= 88
## x= 4 y= 132 first= 22 second= 24
## x= 4 y= 132 first= 33 second= 16
## x= 4 y= 132 first= 12 second= 44
## [1] 1 5 131
## x= 5 y= 131 first= 1 second= 655
## x= 5 y= 131 first= 5 second= 131
## [1] 1 2 2 3 5 13
## x= 6 y= 130 first= 1 second= 780
## x= 6 y= 130 first= 2 second= 390
## x= 6 y= 130 first= 3 second= 260
## x= 6 y= 130 first= 5 second= 156
## x= 6 y= 130 first= 13 second= 60
## x= 6 y= 130 first= 4 second= 195
## x= 6 y= 130 first= 6 second= 130
## x= 6 y= 130 first= 10 second= 78
## x= 6 y= 130 first= 26 second= 30
## x= 6 y= 130 first= 15 second= 52
## x= 6 y= 130 first= 39 second= 20
## x= 6 y= 130 first= 65 second= 12
## [1] 1 3 7 43
## x= 7 y= 129 first= 1 second= 903
## x= 7 y= 129 first= 3 second= 301
## x= 7 y= 129 first= 7 second= 129
## x= 7 y= 129 first= 43 second= 21
## [1] 1 2 2 2 2 2 2 2 2 2 2
## x= 8 y= 128 first= 1 second= 1024
## x= 8 y= 128 first= 2 second= 512
## x= 8 y= 128 first= 4 second= 256
## x= 8 y= 128 first= 8 second= 128
## x= 8 y= 128 first= 32 second= 32
## [1] 1 3 3 127
## x= 9 y= 127 first= 1 second= 1143
## x= 9 y= 127 first= 127 second= 9
## [1] 1 2 2 3 3 5 7
## x= 10 y= 126 first= 1 second= 1260
## x= 10 y= 126 first= 3 second= 420
## x= 10 y= 126 first= 5 second= 252
## x= 10 y= 126 first= 7 second= 180
## x= 10 y= 126 first= 4 second= 315
## x= 10 y= 126 first= 6 second= 210
## x= 10 y= 126 first= 10 second= 126
## x= 10 y= 126 first= 9 second= 140
## x= 10 y= 126 first= 15 second= 84
## x= 10 y= 126 first= 21 second= 60
## x= 10 y= 126 first= 35 second= 36
## x= 10 y= 126 first= 12 second= 105
## x= 10 y= 126 first= 20 second= 63
## x= 10 y= 126 first= 28 second= 45
## x= 10 y= 126 first= 18 second= 70
## [1] 1 5 5 5 11
## x= 11 y= 125 first= 1 second= 1375
## x= 11 y= 125 first= 5 second= 275
## x= 11 y= 125 first= 11 second= 125
## [1] 1 2 2 2 2 3 31
## x= 12 y= 124 first= 1 second= 1488
## x= 12 y= 124 first= 2 second= 744
## x= 12 y= 124 first= 3 second= 496
## x= 12 y= 124 first= 31 second= 48
## x= 12 y= 124 first= 4 second= 372
## x= 12 y= 124 first= 6 second= 248
## x= 12 y= 124 first= 62 second= 24
## x= 12 y= 124 first= 93 second= 16
## x= 12 y= 124 first= 12 second= 124
## [1] 1 3 13 41
## x= 13 y= 123 first= 1 second= 1599
## x= 13 y= 123 first= 3 second= 533
## x= 13 y= 123 first= 13 second= 123
## [1] 1 2 2 7 61
## x= 14 y= 122 first= 1 second= 1708
## x= 14 y= 122 first= 2 second= 854
## x= 14 y= 122 first= 7 second= 244
## x= 14 y= 122 first= 61 second= 28
## x= 14 y= 122 first= 4 second= 427
## x= 14 y= 122 first= 14 second= 122
## [1] 1 3 5 11 11
## x= 15 y= 121 first= 1 second= 1815
## x= 15 y= 121 first= 11 second= 165
## x= 15 y= 121 first= 15 second= 121
## x= 15 y= 121 first= 33 second= 55
## [1] 1 2 2 2 2 2 2 2 3 5
## x= 16 y= 120 first= 1 second= 1920
## x= 16 y= 120 first= 2 second= 960
## x= 16 y= 120 first= 3 second= 640
## x= 16 y= 120 first= 5 second= 384
## x= 16 y= 120 first= 4 second= 480
## x= 16 y= 120 first= 6 second= 320
## x= 16 y= 120 first= 10 second= 192
## x= 16 y= 120 first= 15 second= 128
## x= 16 y= 120 first= 8 second= 240
## x= 16 y= 120 first= 12 second= 160
## x= 16 y= 120 first= 20 second= 96
## x= 16 y= 120 first= 30 second= 64
## x= 16 y= 120 first= 16 second= 120
## x= 16 y= 120 first= 40 second= 48
## x= 16 y= 120 first= 60 second= 32
## [1] 1 7 17 17
## x= 17 y= 119 first= 1 second= 2023
## x= 17 y= 119 first= 7 second= 289
## x= 17 y= 119 first= 17 second= 119
## [1] 1 2 2 3 3 59
## x= 18 y= 118 first= 1 second= 2124
## x= 18 y= 118 first= 3 second= 708
## x= 18 y= 118 first= 59 second= 36
## x= 18 y= 118 first= 4 second= 531
## x= 18 y= 118 first= 118 second= 18
## x= 18 y= 118 first= 9 second= 236
## x= 18 y= 118 first= 177 second= 12
## [1] 1 3 3 13 19
## x= 19 y= 117 first= 1 second= 2223
## x= 19 y= 117 first= 13 second= 171
## x= 19 y= 117 first= 19 second= 117
## x= 19 y= 117 first= 9 second= 247
## x= 19 y= 117 first= 39 second= 57
## [1] 1 2 2 2 2 5 29
## x= 20 y= 116 first= 1 second= 2320
## x= 20 y= 116 first= 2 second= 1160
## x= 20 y= 116 first= 5 second= 464
## x= 20 y= 116 first= 29 second= 80
## x= 20 y= 116 first= 4 second= 580
## x= 20 y= 116 first= 145 second= 16
## x= 20 y= 116 first= 8 second= 290
## x= 20 y= 116 first= 20 second= 116
## [1] 1 3 5 7 23
## x= 21 y= 115 first= 1 second= 2415
## x= 21 y= 115 first= 3 second= 805
## x= 21 y= 115 first= 7 second= 345
## x= 21 y= 115 first= 15 second= 161
## x= 21 y= 115 first= 21 second= 115
## [1] 1 2 2 3 11 19
## x= 22 y= 114 first= 1 second= 2508
## x= 22 y= 114 first= 2 second= 1254
## x= 22 y= 114 first= 3 second= 836
## x= 22 y= 114 first= 11 second= 228
## x= 22 y= 114 first= 19 second= 132
## x= 22 y= 114 first= 4 second= 627
## x= 22 y= 114 first= 6 second= 418
## x= 22 y= 114 first= 22 second= 114
## x= 22 y= 114 first= 33 second= 76
## x= 22 y= 114 first= 57 second= 44
## x= 22 y= 114 first= 209 second= 12
## [1] 1 23 113
## x= 23 y= 113 first= 1 second= 2599
## x= 23 y= 113 first= 23 second= 113
## [1] 1 2 2 2 2 2 2 2 3 7
## x= 24 y= 112 first= 1 second= 2688
## x= 24 y= 112 first= 2 second= 1344
## x= 24 y= 112 first= 3 second= 896
## x= 24 y= 112 first= 7 second= 384
## x= 24 y= 112 first= 4 second= 672
## x= 24 y= 112 first= 6 second= 448
## x= 24 y= 112 first= 14 second= 192
## x= 24 y= 112 first= 21 second= 128
## x= 24 y= 112 first= 8 second= 336
## x= 24 y= 112 first= 12 second= 224
## x= 24 y= 112 first= 28 second= 96
## x= 24 y= 112 first= 42 second= 64
## x= 24 y= 112 first= 16 second= 168
## x= 24 y= 112 first= 24 second= 112
## x= 24 y= 112 first= 84 second= 32
## [1] 1 3 5 5 37
## x= 25 y= 111 first= 1 second= 2775
## x= 25 y= 111 first= 3 second= 925
## x= 25 y= 111 first= 5 second= 555
## x= 25 y= 111 first= 37 second= 75
## x= 25 y= 111 first= 111 second= 25
## [1] 1 2 2 5 11 13
## x= 26 y= 110 first= 1 second= 2860
## x= 26 y= 110 first= 2 second= 1430
## x= 26 y= 110 first= 5 second= 572
## x= 26 y= 110 first= 11 second= 260
## x= 26 y= 110 first= 13 second= 220
## x= 26 y= 110 first= 4 second= 715
## x= 26 y= 110 first= 10 second= 286
## x= 26 y= 110 first= 26 second= 110
## x= 26 y= 110 first= 55 second= 52
## x= 26 y= 110 first= 65 second= 44
## x= 26 y= 110 first= 143 second= 20
## [1] 1 3 3 3 109
## x= 27 y= 109 first= 1 second= 2943
## x= 27 y= 109 first= 109 second= 27
## x= 27 y= 109 first= 9 second= 327
## [1] 1 2 2 2 2 3 3 3 7
## x= 28 y= 108 first= 1 second= 3024
## x= 28 y= 108 first= 2 second= 1512
## x= 28 y= 108 first= 3 second= 1008
## x= 28 y= 108 first= 7 second= 432
## x= 28 y= 108 first= 4 second= 756
## x= 28 y= 108 first= 9 second= 336
## x= 28 y= 108 first= 21 second= 144
## x= 28 y= 108 first= 8 second= 378
## x= 28 y= 108 first= 28 second= 108
## x= 28 y= 108 first= 18 second= 168
## x= 28 y= 108 first= 27 second= 112
## x= 28 y= 108 first= 63 second= 48
## x= 28 y= 108 first= 16 second= 189
## x= 28 y= 108 first= 36 second= 84
## [1] 1 29 107
## x= 29 y= 107 first= 1 second= 3103
## x= 29 y= 107 first= 29 second= 107
## [1] 1 2 2 3 5 53
## x= 30 y= 106 first= 1 second= 3180
## x= 30 y= 106 first= 2 second= 1590
## x= 30 y= 106 first= 3 second= 1060
## x= 30 y= 106 first= 5 second= 636
## x= 30 y= 106 first= 53 second= 60
## x= 30 y= 106 first= 4 second= 795
## x= 30 y= 106 first= 6 second= 530
## x= 30 y= 106 first= 10 second= 318
## x= 30 y= 106 first= 106 second= 30
## x= 30 y= 106 first= 15 second= 212
## x= 30 y= 106 first= 159 second= 20
## x= 30 y= 106 first= 265 second= 12
## [1] 1 3 5 7 31
## x= 31 y= 105 first= 1 second= 3255
## x= 31 y= 105 first= 5 second= 651
## x= 31 y= 105 first= 7 second= 465
## x= 31 y= 105 first= 31 second= 105
## x= 31 y= 105 first= 15 second= 217
## x= 31 y= 105 first= 21 second= 155
## [1] 1 2 2 2 2 2 2 2 2 13
## x= 32 y= 104 first= 1 second= 3328
## x= 32 y= 104 first= 2 second= 1664
## x= 32 y= 104 first= 13 second= 256
## x= 32 y= 104 first= 4 second= 832
## x= 32 y= 104 first= 26 second= 128
## x= 32 y= 104 first= 8 second= 416
## x= 32 y= 104 first= 52 second= 64
## x= 32 y= 104 first= 104 second= 32
## [1] 1 3 11 103
## x= 33 y= 103 first= 1 second= 3399
## x= 33 y= 103 first= 3 second= 1133
## x= 33 y= 103 first= 11 second= 309
## x= 33 y= 103 first= 103 second= 33
## [1] 1 2 2 3 17 17
## x= 34 y= 102 first= 1 second= 3468
## x= 34 y= 102 first= 2 second= 1734
## x= 34 y= 102 first= 3 second= 1156
## x= 34 y= 102 first= 17 second= 204
## x= 34 y= 102 first= 4 second= 867
## x= 34 y= 102 first= 6 second= 578
## x= 34 y= 102 first= 34 second= 102
## x= 34 y= 102 first= 51 second= 68
## x= 34 y= 102 first= 289 second= 12
## [1] 1 5 7 101
## x= 35 y= 101 first= 1 second= 3535
## x= 35 y= 101 first= 5 second= 707
## x= 35 y= 101 first= 7 second= 505
## x= 35 y= 101 first= 101 second= 35
## [1] 1 2 2 2 2 3 3 5 5
## x= 36 y= 100 first= 1 second= 3600
## x= 36 y= 100 first= 3 second= 1200
## x= 36 y= 100 first= 5 second= 720
## x= 36 y= 100 first= 4 second= 900
## x= 36 y= 100 first= 6 second= 600
## x= 36 y= 100 first= 10 second= 360
## x= 36 y= 100 first= 9 second= 400
## x= 36 y= 100 first= 15 second= 240
## x= 36 y= 100 first= 25 second= 144
## x= 36 y= 100 first= 18 second= 200
## x= 36 y= 100 first= 50 second= 72
## x= 36 y= 100 first= 45 second= 80
## x= 36 y= 100 first= 75 second= 48
## x= 36 y= 100 first= 16 second= 225
## x= 36 y= 100 first= 40 second= 90
## x= 36 y= 100 first= 36 second= 100
## x= 36 y= 100 first= 60 second= 60
## [1] 1 3 3 11 37
## x= 37 y= 99 first= 1 second= 3663
## x= 37 y= 99 first= 11 second= 333
## x= 37 y= 99 first= 37 second= 99
## x= 37 y= 99 first= 9 second= 407
## x= 37 y= 99 first= 33 second= 111
## [1] 1 2 2 7 7 19
## x= 38 y= 98 first= 1 second= 3724
## x= 38 y= 98 first= 2 second= 1862
## x= 38 y= 98 first= 7 second= 532
## x= 38 y= 98 first= 19 second= 196
## x= 38 y= 98 first= 4 second= 931
## x= 38 y= 98 first= 14 second= 266
## x= 38 y= 98 first= 38 second= 98
## x= 38 y= 98 first= 49 second= 76
## x= 38 y= 98 first= 133 second= 28
## [1] 1 3 13 97
## x= 39 y= 97 first= 1 second= 3783
## x= 39 y= 97 first= 3 second= 1261
## x= 39 y= 97 first= 13 second= 291
## x= 39 y= 97 first= 97 second= 39
## [1] 1 2 2 2 2 2 2 2 2 3 5
## x= 40 y= 96 first= 1 second= 3840
## x= 40 y= 96 first= 2 second= 1920
## x= 40 y= 96 first= 3 second= 1280
## x= 40 y= 96 first= 5 second= 768
## x= 40 y= 96 first= 4 second= 960
## x= 40 y= 96 first= 6 second= 640
## x= 40 y= 96 first= 10 second= 384
## x= 40 y= 96 first= 15 second= 256
## x= 40 y= 96 first= 16 second= 240
## x= 40 y= 96 first= 24 second= 160
## x= 40 y= 96 first= 40 second= 96
## x= 40 y= 96 first= 60 second= 64
## [1] 1 5 19 41
## x= 41 y= 95 first= 1 second= 3895
## x= 41 y= 95 first= 5 second= 779
## x= 41 y= 95 first= 41 second= 95
## [1] 1 2 2 3 7 47
## x= 42 y= 94 first= 1 second= 3948
## x= 42 y= 94 first= 2 second= 1974
## x= 42 y= 94 first= 3 second= 1316
## x= 42 y= 94 first= 7 second= 564
## x= 42 y= 94 first= 47 second= 84
## x= 42 y= 94 first= 4 second= 987
## x= 42 y= 94 first= 6 second= 658
## x= 42 y= 94 first= 14 second= 282
## x= 42 y= 94 first= 94 second= 42
## x= 42 y= 94 first= 21 second= 188
## x= 42 y= 94 first= 141 second= 28
## x= 42 y= 94 first= 329 second= 12
## [1] 1 3 31 43
## x= 43 y= 93 first= 1 second= 3999
## x= 43 y= 93 first= 3 second= 1333
## x= 43 y= 93 first= 31 second= 129
## x= 43 y= 93 first= 43 second= 93
## [1] 1 2 2 2 2 11 23
## x= 44 y= 92 first= 1 second= 4048
## x= 44 y= 92 first= 2 second= 2024
## x= 44 y= 92 first= 11 second= 368
## x= 44 y= 92 first= 23 second= 176
## x= 44 y= 92 first= 4 second= 1012
## x= 44 y= 92 first= 22 second= 184
## x= 44 y= 92 first= 46 second= 88
## x= 44 y= 92 first= 253 second= 16
## x= 44 y= 92 first= 8 second= 506
## x= 44 y= 92 first= 44 second= 92
## [1] 1 3 3 5 7 13
## x= 45 y= 91 first= 1 second= 4095
## x= 45 y= 91 first= 7 second= 585
## x= 45 y= 91 first= 13 second= 315
## x= 45 y= 91 first= 15 second= 273
## x= 45 y= 91 first= 21 second= 195
## x= 45 y= 91 first= 39 second= 105
## x= 45 y= 91 first= 91 second= 45
## [1] 1 2 2 3 3 5 23
## x= 46 y= 90 first= 1 second= 4140
## x= 46 y= 90 first= 2 second= 2070
## x= 46 y= 90 first= 3 second= 1380
## x= 46 y= 90 first= 5 second= 828
## x= 46 y= 90 first= 23 second= 180
## x= 46 y= 90 first= 4 second= 1035
## x= 46 y= 90 first= 6 second= 690
## x= 46 y= 90 first= 10 second= 414
## x= 46 y= 90 first= 46 second= 90
## x= 46 y= 90 first= 9 second= 460
## x= 46 y= 90 first= 15 second= 276
## x= 46 y= 90 first= 69 second= 60
## x= 46 y= 90 first= 115 second= 36
## x= 46 y= 90 first= 12 second= 345
## x= 46 y= 90 first= 20 second= 207
## x= 46 y= 90 first= 92 second= 45
## x= 46 y= 90 first= 18 second= 230
## [1] 1 47 89
## x= 47 y= 89 first= 1 second= 4183
## x= 47 y= 89 first= 47 second= 89
## [1] 1 2 2 2 2 2 2 2 3 11
## x= 48 y= 88 first= 1 second= 4224
## x= 48 y= 88 first= 3 second= 1408
## x= 48 y= 88 first= 11 second= 384
## x= 48 y= 88 first= 4 second= 1056
## x= 48 y= 88 first= 22 second= 192
## x= 48 y= 88 first= 33 second= 128
## x= 48 y= 88 first= 8 second= 528
## x= 48 y= 88 first= 12 second= 352
## x= 48 y= 88 first= 66 second= 64
## x= 48 y= 88 first= 16 second= 264
## x= 48 y= 88 first= 88 second= 48
## [1] 1 3 7 7 29
## x= 49 y= 87 first= 1 second= 4263
## x= 49 y= 87 first= 7 second= 609
## x= 49 y= 87 first= 29 second= 147
## x= 49 y= 87 first= 87 second= 49
## [1] 1 2 2 5 5 43
## x= 50 y= 86 first= 1 second= 4300
## x= 50 y= 86 first= 2 second= 2150
## x= 50 y= 86 first= 5 second= 860
## x= 50 y= 86 first= 43 second= 100
## x= 50 y= 86 first= 4 second= 1075
## x= 50 y= 86 first= 86 second= 50
## x= 50 y= 86 first= 25 second= 172
## x= 50 y= 86 first= 215 second= 20
## [1] 1 3 5 17 17
## x= 51 y= 85 first= 1 second= 4335
## x= 51 y= 85 first= 5 second= 867
## x= 51 y= 85 first= 15 second= 289
## x= 51 y= 85 first= 51 second= 85
## [1] 1 2 2 2 2 3 7 13
## x= 52 y= 84 first= 1 second= 4368
## x= 52 y= 84 first= 2 second= 2184
## x= 52 y= 84 first= 3 second= 1456
## x= 52 y= 84 first= 7 second= 624
## x= 52 y= 84 first= 13 second= 336
## x= 52 y= 84 first= 4 second= 1092
## x= 52 y= 84 first= 14 second= 312
## x= 52 y= 84 first= 21 second= 208
## x= 52 y= 84 first= 39 second= 112
## x= 52 y= 84 first= 91 second= 48
## x= 52 y= 84 first= 8 second= 546
## x= 52 y= 84 first= 12 second= 364
## x= 52 y= 84 first= 28 second= 156
## x= 52 y= 84 first= 52 second= 84
## x= 52 y= 84 first= 42 second= 104
## x= 52 y= 84 first= 78 second= 56
## x= 52 y= 84 first= 182 second= 24
## x= 52 y= 84 first= 273 second= 16
## [1] 1 53 83
## x= 53 y= 83 first= 1 second= 4399
## x= 53 y= 83 first= 53 second= 83
## [1] 1 2 2 3 3 3 41
## x= 54 y= 82 first= 1 second= 4428
## x= 54 y= 82 first= 2 second= 2214
## x= 54 y= 82 first= 3 second= 1476
## x= 54 y= 82 first= 41 second= 108
## x= 54 y= 82 first= 4 second= 1107
## x= 54 y= 82 first= 82 second= 54
## x= 54 y= 82 first= 9 second= 492
## x= 54 y= 82 first= 123 second= 36
## x= 54 y= 82 first= 12 second= 369
## x= 54 y= 82 first= 164 second= 27
## [1] 1 3 3 3 3 5 11
## x= 55 y= 81 first= 1 second= 4455
## x= 55 y= 81 first= 5 second= 891
## x= 55 y= 81 first= 11 second= 405
## x= 55 y= 81 first= 55 second= 81
## x= 55 y= 81 first= 45 second= 99
## [1] 1 2 2 2 2 2 2 2 5 7
## x= 56 y= 80 first= 1 second= 4480
## x= 56 y= 80 first= 2 second= 2240
## x= 56 y= 80 first= 5 second= 896
## x= 56 y= 80 first= 7 second= 640
## x= 56 y= 80 first= 14 second= 320
## x= 56 y= 80 first= 35 second= 128
## x= 56 y= 80 first= 8 second= 560
## x= 56 y= 80 first= 20 second= 224
## x= 56 y= 80 first= 28 second= 160
## x= 56 y= 80 first= 70 second= 64
## x= 56 y= 80 first= 16 second= 280
## x= 56 y= 80 first= 56 second= 80
## x= 56 y= 80 first= 140 second= 32
## [1] 1 3 19 79
## x= 57 y= 79 first= 1 second= 4503
## x= 57 y= 79 first= 3 second= 1501
## x= 57 y= 79 first= 19 second= 237
## x= 57 y= 79 first= 79 second= 57
## [1] 1 2 2 3 13 29
## x= 58 y= 78 first= 1 second= 4524
## x= 58 y= 78 first= 2 second= 2262
## x= 58 y= 78 first= 3 second= 1508
## x= 58 y= 78 first= 13 second= 348
## x= 58 y= 78 first= 29 second= 156
## x= 58 y= 78 first= 4 second= 1131
## x= 58 y= 78 first= 6 second= 754
## x= 58 y= 78 first= 58 second= 78
## x= 58 y= 78 first= 39 second= 116
## x= 58 y= 78 first= 87 second= 52
## x= 58 y= 78 first= 377 second= 12
## [1] 1 7 11 59
## x= 59 y= 77 first= 1 second= 4543
## x= 59 y= 77 first= 7 second= 649
## x= 59 y= 77 first= 11 second= 413
## x= 59 y= 77 first= 59 second= 77
## [1] 1 2 2 2 2 3 5 19
## x= 60 y= 76 first= 1 second= 4560
## x= 60 y= 76 first= 3 second= 1520
## x= 60 y= 76 first= 5 second= 912
## x= 60 y= 76 first= 19 second= 240
## x= 60 y= 76 first= 4 second= 1140
## x= 60 y= 76 first= 6 second= 760
## x= 60 y= 76 first= 10 second= 456
## x= 60 y= 76 first= 15 second= 304
## x= 60 y= 76 first= 57 second= 80
## x= 60 y= 76 first= 95 second= 48
## x= 60 y= 76 first= 12 second= 380
## x= 60 y= 76 first= 20 second= 228
## x= 60 y= 76 first= 76 second= 60
## x= 60 y= 76 first= 114 second= 40
## x= 60 y= 76 first= 190 second= 24
## x= 60 y= 76 first= 285 second= 16
## [1] 1 3 5 5 61
## x= 61 y= 75 first= 1 second= 4575
## x= 61 y= 75 first= 3 second= 1525
## x= 61 y= 75 first= 61 second= 75
## x= 61 y= 75 first= 15 second= 305
## x= 61 y= 75 first= 183 second= 25
## [1] 1 2 2 31 37
## x= 62 y= 74 first= 1 second= 4588
## x= 62 y= 74 first= 2 second= 2294
## x= 62 y= 74 first= 31 second= 148
## x= 62 y= 74 first= 37 second= 124
## x= 62 y= 74 first= 4 second= 1147
## x= 62 y= 74 first= 62 second= 74
## [1] 1 3 3 7 73
## x= 63 y= 73 first= 1 second= 4599
## x= 63 y= 73 first= 3 second= 1533
## x= 63 y= 73 first= 7 second= 657
## x= 63 y= 73 first= 73 second= 63
## x= 63 y= 73 first= 9 second= 511
## [1] 1 2 2 2 2 2 2 2 2 2 3 3
## x= 64 y= 72 first= 1 second= 4608
## x= 64 y= 72 first= 2 second= 2304
## x= 64 y= 72 first= 3 second= 1536
## x= 64 y= 72 first= 4 second= 1152
## x= 64 y= 72 first= 9 second= 512
## x= 64 y= 72 first= 8 second= 576
## x= 64 y= 72 first= 12 second= 384
## x= 64 y= 72 first= 18 second= 256
## x= 64 y= 72 first= 16 second= 288
## x= 64 y= 72 first= 24 second= 192
## x= 64 y= 72 first= 32 second= 144
## x= 64 y= 72 first= 48 second= 96
## x= 64 y= 72 first= 72 second= 64
## [1] 1 5 13 71
## x= 65 y= 71 first= 1 second= 4615
## x= 65 y= 71 first= 5 second= 923
## x= 65 y= 71 first= 71 second= 65
## [1] 1 2 2 3 5 7 11
## x= 66 y= 70 first= 1 second= 4620
## x= 66 y= 70 first= 3 second= 1540
## x= 66 y= 70 first= 5 second= 924
## x= 66 y= 70 first= 7 second= 660
## x= 66 y= 70 first= 11 second= 420
## x= 66 y= 70 first= 4 second= 1155
## x= 66 y= 70 first= 6 second= 770
## x= 66 y= 70 first= 10 second= 462
## x= 66 y= 70 first= 14 second= 330
## x= 66 y= 70 first= 22 second= 210
## x= 66 y= 70 first= 15 second= 308
## x= 66 y= 70 first= 21 second= 220
## x= 66 y= 70 first= 33 second= 140
## x= 66 y= 70 first= 35 second= 132
## x= 66 y= 70 first= 55 second= 84
## x= 66 y= 70 first= 77 second= 60
## x= 66 y= 70 first= 12 second= 385
## x= 66 y= 70 first= 20 second= 231
## x= 66 y= 70 first= 28 second= 165
## x= 66 y= 70 first= 44 second= 105
## x= 66 y= 70 first= 30 second= 154
## x= 66 y= 70 first= 66 second= 70
## [1] 1 3 23 67
## x= 67 y= 69 first= 1 second= 4623
## x= 67 y= 69 first= 67 second= 69
## [1] 1 2 2 2 2 17 17
## x= 68 y= 68 first= 1 second= 4624
## x= 68 y= 68 first= 2 second= 2312
## x= 68 y= 68 first= 17 second= 272
## x= 68 y= 68 first= 4 second= 1156
## x= 68 y= 68 first= 34 second= 136
## x= 68 y= 68 first= 289 second= 16
## x= 68 y= 68 first= 8 second= 578
## x= 68 y= 68 first= 68 second= 68