1
                                   /
                                  |      1
        On the Irrationality of,  |   -------- dx, for a from 1 to , 40
                                  |         5
                                  |        x
                                 /    1 + ----
                                   0       a



                               By Shalosh B. Ekhad



In this computer-generated book, accompanying the article by Doron Zeilberge\
    r and Wadim Zudilin

 "Towards Automatic Discovery of Irrationailty Proofs and Irrationality Meas\
    ures."



 I will automatically prove irrationality, and establish irrationality measu\
    res, for the constant



                                   1
                                  /
                                 |      1
                                 |   -------- dx
                                 |         5
                                 |        x
                                /    1 + ----
                                  0       a



                              for a from 1 to, 40



           I will also state the cases where we were unable to do it.



                                                                   1
                                                                  /
                                                                 |     1
 We are unable, with this method to prove the irrationality of,  |   ------ dx
                                                                 |    5
                                                                /    x  + 1
                                                                  0

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       2

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       3

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       4

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       5

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       6

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       7

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       8

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       9

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.1075967704

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       10

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       11

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       12

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       13

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       14

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.1629529738

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       15

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       16

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       17

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       18

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       19

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.1963894741

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       20

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       21

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       22

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       23

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.001417476473

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       24

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.2815216531

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       25

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.005709844206

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       26

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.01336710794

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       27

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.01947684981

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       28

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.02328678786

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       29

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.2393294020

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       30

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.02921475867

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       31

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.03212734020

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       32

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.03547970100

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       33

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.03827152558

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       34

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.2568568411

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       35

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.04376413892

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       36

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.05234801788

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       37

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.04903733173

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       38

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.05503587845

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       39

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.2693209901

                                                                  1
                                                                 /
                                                                |      1
We are unable, with this method to prove the irrationality of,  |   -------- dx,
                                                                |         5
                                                                |        x
                                                               /    1 + ----
                                                                 0       40

    but it seems that a sharperning of

                  the divisibility lemma might lead to a proof

since the smallest empirical delta from, 100, to , 200,  is positive:,

    0.05762730337



                -----------------------------------------------



         This ends this book that took, 2122.483, seconds to generate.