decimal expression of mersenne candidates

[m_f_h]

Hi,
I noticed that most of the exponents I get to test from primenet look quite similar in their last decimal places. I mean, it's not surprising that the last digit is always 3,7 or 9, but :
* in almost half the cases, the second last digit is 7
* several (I did not yet test hundreds...) end up in ...773
* many end up in -23 or -83

Is this just a cioncidence ? Or is there at least a partial explanation ?
Since I don't know a reason for which particular endings should be favourite among prime numbers.
Does anyone have an idea on that ?

[MooMoo2]

Originally Posted by m_f_h

Hi,
I noticed that most of the exponents I get to test from primenet look quite similar in their last decimal places. I mean, it's not surprising that the last digit is always 3,7 or 9,

The last digit is not always 3, 7, or 9. The prime 2^25964951-1 has an exponent that ends in 1, while the last few digits of GIMPS's last prime is:

053967871

[cheesehead]

Originally Posted by m_f_h

* in almost half the cases, the second last digit is 7
* several (I did not yet test hundreds...) end up in ...773
* many end up in -23 or -83

Is this just a cioncidence ?

It's just normal statistical variation.

Since I don't know a reason for which particular endings should be favourite among prime numbers.
Does anyone have an idea on that ?

The ending digits will be partially determined by the prime divisors of the base in which the number is expressed.

Since 10 is divisible by 2 and 5, any base-10 number greater than 5 will be composite if its final digit is 2, 4, 6, 8 (because of 2-divisibility), 5 (because of 5-divisibility), or 0 (because of combined 2- and 5-divisibility). So any prime greater than 5 must end in 1, 3, 7, or 9.

Since 2 and 5 are the only prime divisors of 10, we can't make any further final-digit rules.

Now, if instead you were to express the numbers in base-30 rather than base-10, you could use not only the prime divisors 2 and 5 of 30, but also the divisor 3, to construct rules for the final (base-30-)digit of primes versus composites. You'd find that, in base-30, the final (base-30-)digit of a prime (greater than 5, that is) could be only 1, 7, 11, 13, 17, 19, 23, or 29. Any base-30 number with a final (base-30-)digit of 9, for instance, would be composite, guaranteed to be divisible by 3 because it could be expressed as 30k+9 for some integer k.

[ATH]

Primenet gives out exponent in the 35M range now. Testing the occurence of the 2-digit ending of the 57,487 primes between 35,000,000 and 36,000,000 gives:
01: 1442
03: 1448
07: 1438
09: 1438
11: 1421
13: 1455
17: 1481
19: 1418
21: 1420
23: 1464
27: 1443
29: 1458
31: 1454
33: 1424
37: 1408
39: 1452
41: 1419
43: 1409
47: 1445
49: 1451
51: 1428
53: 1414
57: 1422
59: 1404
61: 1422
63: 1455
67: 1425
69: 1457
71: 1402
73: 1401
77: 1453
79: 1435
81: 1439
83: 1506
87: 1445
89: 1439
91: 1435
93: 1448
97: 1438
99: 1431

So there is no significant variation.

[ATH]

After removing the 35,828 primes between 35M and 36M where factors have been found (factors.zip), the last 2 digits of the remaining 21,659 candidates released by primenet are still pretty evenly distributed:

01: 537, 03: 516, 07: 569, 09: 563, 11: 525, 13: 554, 17: 589, 19: 508, 21: 539, 23: 551
27: 560, 29: 593, 31: 510, 33: 517, 37: 518, 39: 509, 41: 562, 43: 506, 47: 588, 49: 573
51: 510, 53: 541, 57: 521, 59: 496, 61: 545, 63: 513, 67: 557, 69: 579, 71: 537, 73: 527
77: 574, 79: 486, 81: 543, 83: 562, 87: 554, 89: 599, 91: 516, 93: 558, 97: 529, 99: 525

[m_f_h]

Originally Posted by ATH

After removing the 35,828 primes between 35M and 36M where factors have been found (factors.zip), the last 2 digits of the remaining 21,659 candidates released by primenet are still pretty evenly distributed:
(...)

OK, +1 for you. In the meantime I wrote a quick maple hack that showed about the same for arbitrary ranges.

This issue popped into my eyes when I got 36846773 just after 36279773, and since I also have/had
36188623, 37478923, 36832483, 37567583 among the few (<20) I've got so far.

So it's just another variation on the law of big numbers....

As to the previous replies:
sorry for the "1" I forgot, last digits are of course in {1,3,7,9}, this is so to say trivial (must be odd and not multiples of 5).
But, as you noticed, no (easy to see) rule can't be made that goes beyond the very last digit ; that was the issue I was talking about.

[m_f_h]

Your numbers give the following distribution by "decade":
[48] => ,79
[49] => ,59
[50] => ,19,39,43
[51] => ,3,31,33,37,51,63,91
[52] => ,11,57,73,97,99
[53] => ,1,21,71
[54] => ,53,61,81
[55] => ,13,23,67,87,93
[56] => ,7,9,27,41,83
[57] => ,49,69,77
[58] => ,17,47
[59] => ,29,89

it's not really a Gaussian... if someone wants to do that for the different categories (persenne primes, their exponents, "candidates", a "pre-sieved" range in M30-M40...) here's the code:
$a=eval(str_replace(":","=>","return array(
01: 537, 03: 516, 07: 569, 9: 563, 11: 525, 13: 554, 17: 589, 19: 508, 21: 539, 23: 551,
27: 560, 29: 593, 31: 510, 33: 517, 37: 518, 39: 509, 41: 562, 43: 506, 47: 588, 49: 573,
51: 510, 53: 541, 57: 521, 59: 496, 61: 545, 63: 513, 67: 557, 69: 579, 71: 537, 73: 527,
77: 574, 79: 486, 81: 543, 83: 562, 87: 554, 89: 599, 91: 516, 93: 558, 97: 529, 99: 525
);")); foreach($a as $p=>$f) $b[(int)($f/10)].=",$p"; ksort($b);echo"<pre>",print_r($b);