A first digit theorem for powerful integer powers
- Werner Hürlimann^{1}Email author
Received: 11 August 2015
Accepted: 24 September 2015
Published: 6 October 2015
Abstract
For any fixed power exponent, it is shown that the first digits of powerful integer powers follow a generalized Benford law (GBL) with size-dependent exponent that converges asymptotically to a GBL with the inverse double power exponent. In particular, asymptotically as the power goes to infinity these sequences obey Benford’s law. Moreover, the existence of a one-parametric size-dependent exponent function that converges to these GBL’s is established, and an optimal value that minimizes its deviation to two minimum estimators of the size-dependent exponent is determined. The latter is undertaken over the finite range of powerful integer powers less than \( 10^{s \cdot m} , \, m = 8, \ldots ,15 \), where \( s = 1,2,3,4,5 \) is a fixed power exponent.
Keywords
Mathematics Subject Classification
Background
Clearly, the limiting case \( \alpha \to 0 \), respectively \( \alpha = 1 \), of (2) converges weakly to Benford’s law (1), respectively the uniform distribution. It is expected that many further integer power sequences follow a GBL with size-dependent parameter. However, if asymptotically such an exponent exists, it will not always be exactly \( \alpha = s^{ - 1} \). Hürlimann (2014b) shows that the first digits of powers from perfect power numbers follow asymptotically a GBL with parameter \( \alpha = (2s)^{ - 1} \).
Based on similar statistical analysis, the first digits of powerful integer powers are studied. For this, the GBL is fitted to samples of first digits using two goodness-of-fit measures, namely the MAD measure (mean absolute deviation) and the WLS measure (probability weighted least square or Chi square statistics). In “Size-dependent generalized Benford law for powerful integer powers”, the minimum MAD and WLS estimators of the GBL over finite ranges of powerful integer powers up to \( 10^{s \cdot m} , \, m \ge 8 \), \( s \ge 1 \) a fixed power exponent, are determined. Calculations illustrate the convergence of the size-dependent GBL with minimum MAD and WLS estimators to the GBL with exponent \( (2s)^{ - 1} \). Moreover, the existence of a one-parametric size-dependent exponent function that converges to these GBL’s is established, and an optimal value that minimizes its absolute deviation to the minimum MAD and WLS estimators is determined. A mathematical proof of the asymptotic convergence of the finite sequences to the GBL with exponent \( (2s)^{ - 1} \) follows in “Asymptotic counting function for powerful integer powers”.
Size-dependent generalized Benford law for powerful integer powers
GBL fit for first digit of powerful integer powers: MAD vs. WLS criterion
s = 1 | Parameters | Δ to LL estimate | MAD GoF measures | WLS GoF measures | ||||||
---|---|---|---|---|---|---|---|---|---|---|
m = | WLS | MAD | WLS | MAD | LL | WLS | MAD | LL | WLS | MAD |
8 | 0.50630179 | 0.50602471 | 1.819 | 1.947 | 282.9 | 135.2 | 134.6 | 90.14 | 28.44 | 28.75 |
9 | 0.50398677 | 0.50439385 | 2.298 | 1.890 | 168.3 | 90.24 | 86.04 | 66.89 | 21.19 | 22.62 |
10 | 0.50258542 | 0.50277813 | 2.755 | 2.340 | 92.13 | 45.33 | 45.02 | 42.88 | 12.36 | 13.05 |
11 | 0.50170493 | 0.50180536 | 3.116 | 2.650 | 44.37 | 17.57 | 16.34 | 22.63 | 4.505 | 4.911 |
12 | 0.50116789 | 0.50121344 | 2.932 | 2.477 | 19.77 | 8.163 | 7.927 | 8.998 | 1.546 | 1.726 |
13 | 0.50080206 | 0.50083278 | 2.077 | 1.415 | 6.993 | 4.335 | 4.077 | 2.519 | 0.784 | 0.960 |
14 | 0.50054625 | 0.50054722 | 0.289 | 0.244 | 1.821 | 1.802 | 1.794 | 0.345 | 0.330 | 0.330 |
15 | 0.50037082 | 0.50036094 | 3.110 | 2.122 | 2.042 | 1.252 | 1.200 | 1.154 | 0.315 | 0.400 |
s = 2 | Parameters | Δ to LL estimate | MAD GoF measures | WLS GoF measures | ||||||
---|---|---|---|---|---|---|---|---|---|---|
m = | WLS | MAD | WLS | MAD | LL | WLS | MAD | LL | WLS | MAD |
8 | 0.25428436 | 0.25324296 | 0.383 | 0.867 | 238.4 | 218.8 | 193.9 | 163.9 | 157.8 | 167.6 |
9 | 0.25428436 | 0.25267279 | 1.142 | 0.469 | 76.66 | 122.8 | 59.30 | 28.98 | 66.02 | 28.52 |
10 | 0.25166049 | 0.25155591 | 0.585 | 0.811 | 26.64 | 19.31 | 18.58 | 7.514 | 4.409 | 4.870 |
11 | 0.25096820 | 0.25089373 | 1.021 | 1.366 | 22.15 | 15.50 | 13.56 | 10.95 | 6.568 | 7.071 |
12 | 0.25062939 | 0.25059921 | 1.012 | 1.314 | 11.19 | 8.423 | 7.972 | 6.168 | 4.169 | 4.347 |
13 | 0.25042363 | 0.25041307 | 0.552 | 0.779 | 3.923 | 3.055 | 2.805 | 1.563 | 1.287 | 1.334 |
14 | 0.25028419 | 0.25028723 | 0.369 | 0.510 | 1.533 | 1.302 | 1.238 | 0.540 | 0.482 | 0.491 |
15 | 0.25018970 | 0.25019163 | 1.984 | 2.177 | 1.344 | 0.640 | 0.624 | 1.027 | 0.258 | 0.266 |
s = 3 | Parameters | Δ to LL estimate | MAD GoF measures | WLS GoF measures | ||||||
---|---|---|---|---|---|---|---|---|---|---|
m = | WLS | MAD | WLS | MAD | LL | WLS | MAD | LL | WLS | MAD |
8 | 0.16900766 | 0.16723059 | 0.495 | 1.319 | 274.3 | 257.7 | 248.0 | 575.4 | 553.1 | 615.3 |
9 | 0.16801302 | 0.16762481 | 0.748 | 1.137 | 116.4 | 89.18 | 81.67 | 142.8 | 119.1 | 125.5 |
10 | 0.16769485 | 0.16753300 | 0.560 | 0.909 | 60.96 | 58.26 | 57.25 | 124.0 | 117.8 | 120.2 |
11 | 0.16729754 | 0.16730322 | 0.748 | 0.722 | 18.31 | 13.37 | 13.25 | 24.45 | 19.34 | 19.35 |
12 | 0.16710133 | 0.16708812 | 0.524 | 0.656 | 7.100 | 5.652 | 5.371 | 6.058 | 4.895 | 4.969 |
13 | 0.16695408 | 0.16695069 | 0.260 | 0.333 | 3.798 | 3.374 | 3.255 | 3.664 | 3.530 | 3.541 |
14 | 0.16685701 | 0.16685618 | 0.287 | 0.249 | 2.484 | 2.358 | 2.338 | 3.931 | 3.856 | 3.857 |
15 | 0.16679314 | 0.16679442 | 1.323 | 1.452 | 1.321 | 0.972 | 0.952 | 2.021 | 1.279 | 1.286 |
s = 4 | Parameters | Δ to LL estimate | MAD GoF measures | WLS GoF measures | ||||||
---|---|---|---|---|---|---|---|---|---|---|
m = | WLS | MAD | WLS | MAD | LL | WLS | MAD | LL | WLS | MAD |
8 | 0.12774699 | 0.12760514 | 0.089 | 0.023 | 336.0 | 339.0 | 334.9 | 1704 | 1702 | 1703 |
9 | 0.12682400 | 0.12682674 | 0.253 | 0.256 | 108.7 | 100.3 | 100.2 | 495.4 | 489.5 | 489.6 |
10 | 0.12592594 | 0.12589200 | 0.086 | 0.160 | 38.75 | 37.58 | 36.59 | 149.2 | 148.9 | 149.1 |
11 | 0.12543815 | 0.12539409 | 0.724 | 0.928 | 21.98 | 16.83 | 15.38 | 52.95 | 42.62 | 43.45 |
12 | 0.12530767 | 0.12526505 | 0.576 | 1.002 | 7.884 | 6.275 | 5.841 | 18.01 | 14.97 | 16.63 |
13 | 0.12521961 | 0.12520948 | 0.108 | 0.326 | 4.885 | 4.688 | 4.291 | 20.84 | 20.79 | 20.99 |
14 | 0.12514734 | 0.12514391 | 0.428 | 0.269 | 1.315 | 1.279 | 1.181 | 4.693 | 4.331 | 4.381 |
15 | 0.12509739 | 0.12509771 | 1.246 | 1.278 | 0.940 | 0.587 | 0.579 | 3.039 | 1.619 | 1.620 |
s = 5 | Parameters | Δ to LL estimate | MAD GoF measures | WLS GoF measures | ||||||
---|---|---|---|---|---|---|---|---|---|---|
m = | WLS | MAD | WLS | MAD | LL | WLS | MAD | LL | WLS | MAD |
8 | 0.10483305 | 0.10494392 | 1.295 | 1.346 | 551.6 | 498.4 | 496.2 | 8531 | 7818 | 7819 |
9 | 0.10110871 | 0.10166842 | 0.148 | 0.412 | 182.7 | 185.4 | 175.0 | 2055 | 2051 | 2113 |
10 | 0.10041545 | 0.10050001 | 0.770 | 0.588 | 39.74 | 30.51 | 28.02 | 211.5 | 157.1 | 160.2 |
11 | 0.10035943 | 0.10038750 | 0.538 | 0.407 | 21.67 | 19.45 | 18.85 | 124.1 | 111.8 | 112.5 |
12 | 0.10021129 | 0.10018888 | 0.809 | 1.033 | 10.05 | 8.099 | 7.558 | 60.76 | 47.82 | 48.81 |
13 | 0.10015612 | 0.10013482 | 0.508 | 0.967 | 5.373 | 4.752 | 4.238 | 35.74 | 33.38 | 35.31 |
14 | 0.10011058 | 0.10010322 | 0.004 | 0.337 | 1.790 | 1.792 | 1.653 | 9.827 | 9.827 | 10.32 |
15 | 0.10007403 | 0.10007147 | 0.609 | 0.353 | 0.794 | 0.713 | 0.655 | 4.477 | 3.745 | 3.874 |
Table 1 displays exact results obtained on a computer with single precision, i.e., with 15 significant digits. The MAD (resp. WLS) measures are given in units of \( 10^{ - 6} \) (resp. \( \sqrt[3]{{10^{ - (m + s + 12)} }} \)). Taking into account the decreasing units, one observes that the optimal MAD and WLS measures decrease with increasing sample size.
Asymptotic counting function for powerful integer powers
Therefore, the size-dependent exponent (10) with \( c = 1 \) not only minimizes the absolute deviations between the LL estimator and the MAD (resp. WLS) estimators over the finite ranges of powerful powers \( [ 1 ,10^{s \cdot m} ] , \, m = 8, \ldots ,15, \, s = 1,2,3,4,5 \), as shown in “Size-dependent generalized Benford law for powerful integer powers”, but it turns out to be uniformly best with maximum error less than \( 7.3 \cdot 10^{ - 3} \) against the asymptotic estimate, at least if \( N \ge 10^{16} \). Moreover, the following limiting asymptotic result has been shown.
First digit theorem for powerful integer powers (GBL for powerful integer powers)
Comparison of powerful number counting functions for \( N = 10^{m} \)
m | S (N) lower bound (14) | Q (N) | S (N)/q √N | Q (N)/q √N |
---|---|---|---|---|
8 | 20,880 | 25,708 | 0.960773 | 1.182929 |
9 | 66,888 | 77,311 | 0.973282 | 1.124946 |
10 | 213,371 | 235,726 | 0.981806 | 1.08467 |
11 | 678,723 | 726,421 | 0.987604 | 1.057009 |
12 | 2,154,897 | 2,256,191 | 0.991555 | 1.038165 |
13 | 6,832,881 | 7,047,093 | 0.994247 | 1.025417 |
14 | 21,647,316 | 22,098,596 | 0.99608 | 1.016846 |
15 | 68,540,677 | 69,488,109 | 0.99733 | 1.011116 |
16 | 216,929,613 | 218,912,528 | 0.998181 | 1.007305 |
17 | 686,390,158 | 690,528,822 | 0.998761 | 1.004783 |
18 | 2,171,414,780 | 2,180,031,831 | 0.999156 | 1.003121 |
19 | 6,868,466,063 | 6,886,369,416 | 0.999425 | 1.00203 |
20 | 21,723,981,663 | 21,761,110,713 | 0.999608 | 1.001316 |
21 | 68,705,847,049 | 68,782,727,400 | 0.999733 | 1.000852 |
22 | 217,285,461,383 | 217,444,443,741 | 0.999818 | 1.00055 |
23 | 687,156,809,129 | 687,485,219,306 | 0.999876 | 1.000354 |
24 | 2,173,066,478,000 | 2,173,744,295,846 | 0.999916 | 1.000227 |
25 | 6,872,024,538,797 | 6,873,422,599,112 | 0.999942 | 1.000146 |
Conclusions
The first digits of some integer sequences like integer powers and square-free integer powers to a fixed power exponent \( s \ge 1 \), follow a generalized Benford law with size-dependent exponent that converges asymptotically to a GBL with the inverse power exponent \( s^{ - 1} \). In contrast to this, there exist integer sequences, for which such power sequences behave like a GBL with parameter different from \( s^{ - 1} \). The analysed powerful integer power sequences follow a GBL with parameter \( (2s)^{ - 1} \) and are in this respect similar to powers from perfect power numbers studied previously in Hürlimann (2014b). Moreover, all these power sequences typically are not exactly Benford distributed. Departures from Benford’s law occur quite frequently within mathematics and in almost all related scientific disciplines, and must be analysed along the line of appropriate probabilistic models. The companion papers Hürlimann (2015a, b) introduce and discuss some possibilities for a scientific wider use.
Declarations
Acknowledgements
None to be declared.
Compliance with ethical guidelines
Competing interests The author declares that he has no competing interests.
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Authors’ Affiliations
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