Are the rightmost digits of G random?




Some statistics computed using Fourmilab's ENT test program

Rightmost 10000 decimal digits:

    entropy: 3.3216 bits per digit (cf. 3.3219... = log2(10) for an infinite random sequence) 
    chi-square p-value*: 0.8285 (cf. ≥ 0.01 if not significantly different from random)
    serial correlation coefficient:  -0.004062 (cf. 0 for an infinite random sequence)
    relative digit-frequencies: [0.1032, 0.1007, 0.0972, 0.1014, 0.0979, 0.1016, 0.0999, 0.09970, 0.1025, 0.0959] (cf. all 1/10 for an infinite random sequence) 
    * This chi-square result was computed with custom software, since ENT assumes sampling from 256 byte-values (instead of only ten). 

Rightmost 10000 decimal digits converted to binary (33216 bits):

    entropy: 0.99994 bits per bit (cf. 1 for an infinite random sequence) 
    chi-square p-value: 0.0975 (cf. ≥ 0.01 if not significantly different from random) 
    serial correlation coefficient: -0.004057 (cf. 0 for an infinite random sequence)
    relative digit-frequencies: [0.5045, 0.4955] (cf. both 1/2 for an infinite random sequence)



Some statistics computed using the NIST Statistical Test Suite 

Rightmost 20000 binary digits of G:
    Test                                Result (p-values)           Block length
    ---------------------------------   -------------------------   --------------
    Frequency (Monobit)                 PASS (0.213313)	
    Frequency Test within a Block       PASS (0.631448)             M=200
    Runs                                PASS (0.504183)	
    Longest-Run-of-Ones in a Block      PASS (0.593697)	            M=128 (preset)
    Non-overlapping Template Matching   PASS* 146 of 148 patterns   m=10
    Serial                              PASS (0.593483, 0.619471)   m=10
    Approximate Entropy                 PASS (0.488289)             m=9
    Cumulative Sums                     PASS (0.349060, 0.107114)
                                        PASS means the p-value is at least 0.01
*cf. the rightmost 20000 bits of pi PASS 147 of the 148 patterns in this test.


Tentative conclusion

    Treated simply as test data, these digit strings do not exhibit significant departures from randomness (as far as these few tests are concerned).  
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