What Are Cole Semiprimes?

Cole's multiplication was important because he had succeeded in factorizing the number, 2 to the 67^th power minus 1. This mathematical feat was special since the French monk, Marin Mersenne, had claimed in his book Cogitata Physico-Mathematica (published in 1644) that 2⁶⁷ -1 was a prime number. (In fact, he claimed that if p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, then 2^p-1 is prime - primes of this form have now been dubbed 'Mersenne primes,' and the currently largest known one is 2^32582657-1, a Mersenne prime with 9,808,358 digits.) Although it had been known since 1871 that 2⁶⁷-1 was not prime, (number theorists had used Fermat's little theorem to test it, which says that if a^p = a (mod p) for any integer a, then p is a prime number) no one knew the actual factorization of 2⁶⁷ - 1.

But then on one cold rainy October afternoon in 1903, Frank Cole was scheduled to give a talk at a meeting of the American Mathematical Society with the unassuming title, "On the Factorization of Large Numbers." When his time came, Cole strode confidently to the blackboard and carefully wrote out:

2⁶⁷-1 = 147573952589676412927

Then he moved to another section of the board and began the long process of multiplying two large numbers together: 193707721 and 761838257287.

Cole never spoke a word during the tedious multiplication process. But when he finally laid down his chalk and the product agreed with the original decimal expansion of 2⁶⁷-1:

2⁶⁷-1 = 147573952589676412927

193707721 * 761838257287 = 147573952589676412927

The crowd recognized his accomplishment and broke out in applause. At the meeting, no one asked Cole a single question about his "talk," but later he stated that it had required 20 years of continuous Sunday afternoons to find the factorization of 2⁶⁷ - 1. (It takes my computer with a Pentium 4 processor exactly 125 milliseconds.)

Frank Nelson Cole's factorization accomplishment is now so well-known that an account of it usually makes an appearance in every modern number theory book written for non-mathematicians.

Now that we are familiar with Cole's famous story, let's have a closer look at the factorization of 2⁶⁷ -1. First, we notice that the number is a semiprime (a number having only two prime factors). We could easily search via computer for all semiprimes of the form 2ⁿ -1 and call them Cole semiprimes. Here are the values of n such that 2ⁿ -1 is a semiprime:

4, 9, 11, 23, 37, 41, 49, 59, 67, 83, 97, 101, 103, 109, 131, 137, 139, 149, 167, 197, 199, 227, 241, 269, 271, 281, 293, 347, 373, 379, 421, 457, 487, 523, 727, 809, 881, 971, 983, 997 ...

This is sequence A085724 in the Online Encyclopedia of Integer Sequences.

But if we look at the factors of 2⁶⁷-1 again, we notice that its smallest prime factor has 9 decimal digits while its largest has 12 - a difference of 3 digits. Taking inspiration from this, we could also make a sequence of semiprimes such that the smallest prime factor is 3 digits less than the largest prime factor. But a computer search brings up many terms fitting this definition. So to make the terms more rare (and the sequence perhaps more interesting) we could restrict ourselves to looking for the first semiprime whose smallest prime factor has exactly n digits and whose largest prime factor is 3 digits greater than that.

But a little thought reveals that this sequence has one major problem: It won't include the original Cole semiprime, 2⁶⁷ - 1. Because what the sequence will amount to is finding semiprimes with the form (nextprime greater than 10ⁿ) * (nextprime greater than 10ⁿ), which can be seen in the following table.

n X=Nextprime(10ⁿ) Y=Nextprime(10ⁿ) Semi=X*Y

0 2 1009 2018

1 11 10007 110077

2 101 100003 10100303

3 1009 1000003 1009003027

4 10007 10000019 100070190133

5 100003 100000007 10000300700021

6 1000003 1000000007 1000003007000021

7 10000019 10000000019 100000190190000361

So this isn't really a very interesting sequence. Nevertheless, I have dubbed it "Cole Semiprimes v. 2," the terms of which appear in the fourth column above; and while real mathematicians will probably frown upon seeing the sequence and how it is defined, I still think it is sort of charming in its own way.

Here are four more facts about Frank Nelson Cole:

1. He only published around 25 research papers in his lifetime, which is a relatively modest amount for a mathematician.

2. He was considered an inspiring lecturer by his students and colleagues and frequently incorporated the "hottest" research topics into his talks to make them more interesting and entertaining.

3. Many people believe the level of mathematical instruction increased greatly while Cole was lecturing and tutoring at Harvard University from 1885 to 1888.

4. Cole attained a professorship at Columbia in 1895 and held the post for the rest of his life.

Now I will close with a weird prime curiosity related to Cole's factorization of 2⁶⁷ - 1. Consider a number formed by concatenating the numbers 193707721, 2⁸²³³, and 761838257287, each separated by three zeros. Note that the first and third numbers are the factors that Cole found in 1903. This number is a prime with 2,506 digits and has the following decimal expansion.

193707 7210002398 5805161770 8242926845 9608931767 1017372472
6668424473 0470783245 3250117486 8251165663 0489156905 0709109356 0647473419
4443278670 0100543721 9966903374 9379853240 7120341777 8021151823 6154265256
2732763049 3306697469 6724948558 9502287220 3753445906 9710705829 0313500581
3406475847 7677349658 2055692650 1486143072 3336841932 4143691462 8141273805
8604534943 2770814429 1787225669 8749112627 0620806101 9272190583 7946127268
5870039038 7072054613 9641961775 5830605713 2717430302 6029321186 3222375658
5167395425 3231709751 4004415426 9252372908 8803953952 7336297897 9744565971
8079905160 4361600425 5242268165 9422167268 3673305744 4174303415 4113585286
9512610509 5039775903 8691115652 9663247928 7839910675 6959636003 3680445910
6631018895 7777757104 5204752348 9413493554 9633426611 7857284684 3932450507
9218938564 9826711324 6945573052 8522498302 1620555924 8945681682 0336313652
6186473695 7372469762 8564396858 3129966127 8654536812 4260472173 8525977104
9246036151 9170948428 7305643668 4478304989 7666669435 9604480310 7275322155
8482736200 2343325233 3906220573 0718230546 0322186580 3927752095 5961033619
5652229572 4083690521 1707803913 6182160313 3251802917 6568066442 7932047326
8701902315 3489273402 8065433613 3800145139 1291319133 6516416499 8320229964
2286914323 1170654279 2348896889 7407114562 3439310425 2633171482 4904607905
1506962840 1789341356 5532622392 8418680599 2153229793 5544552447 0315689866
2887758114 3635489197 5844977093 1282137000 1742771586 6174071320 2284324855
4531800364 3515455872 9044534806 5754306774 1153266024 4766090540 7134416244
4025560472 0655421637 9810723340 9401754316 3808626730 1663738564 4611201259
5541539649 0605793416 8450026140 2772861629 2521439752 4189546867 6912982506
6566750796 9255389353 9523205188 9163116753 7572706141 5423050421 7096771474
8541040633 3918399120 4499831648 5763656259 0523206083 6062857201 7855097528
2853499778 6817208304 6026363142 9649701312 7702645821 5313524598 2927011381
0126050173 1254527006 5255077562 6207413501 1691473903 5875831312 4603016682
8613993058 3201619216 6842950150 7701608000 7623184642 6995149332 3430419736
4053515407 6408269775 8885599870 6180744619 3089339063 3030664907 8190510622
0716357534 2979413477 1753824189 0989913447 2832077510 8129858520 1981985146
5708356564 0027681695 2584884887 8530810869 7753700560 4318475032 1776684629
9212152737 8022949547 4058890968 5734807279 7782461790 2541303288 3326227360
9291212953 3425357319 4732245904 8526129280 1511384167 8439484606 3161701096
8025330801 7922896257 3442144284 6520476725 7092967473 2029700263 8787534565
0529032902 1276064419 3428438478 2224494616 9971860269 3181346208 3616392763
5890739300 2833579255 0861725995 3017300793 6916629141 5859200076 1838257287

Awesome.