Posted by Jason Polak on 07. April 2018 · Write a comment · Categories: number-theory · Tags: ,

There is a cool way to express 1 as a sum of unit fractions using partitions of a fixed positive integer. What do we mean by partition? If $n$ is such an integer then a partition is just a sum $e_1d_1 + \cdots + e_kd_k = n$ where $d_i$ are positive integers. For example,

7 = 4 + 1 + 1 = 4 + 2(1)

The order of the partition is not very interesting and so we identify partitions up to order. Thus, here are all the 15 partitions of 7:

7
6+1
5+2
5+1+1
4+3
4+2+1
4+1+1+1
3+3+1
3+2+2
3+2+1+1
3+1+1+1+1
2+2+2+1
2+2+1+1+1
2+1+1+1+1+1
1+1+1+1+1+1+1

Group the same numbers together so that each partition is written as $n = \sum e_id_i$ where there are $e_i$ appearances of $d_i$ (or vice-versa, it's symmetric). Then it's a theorem that:
$$
1 = \sum (e_1!\cdots e_k!\cdot d_1^{e_1}\cdots d_k^{e_k})^{-1}.$$
This partition identity has a bunch of proofs. A neat one appears in the paper "Using Factorizations to Prove a Partition Identity" by David Dobbs and Timothy Kilbourn. In their proof, they used an asympotitc expression for the number of irreducible polynomials over a finite field of a given degree $n$ (the same $n$ that appears in the partition).

Here are some examples of this identity. For n=5, we have:

1 = 1/5 + 1/4 + 1/6 + 1/6 + 1/8 + 1/12 + 1/120

For n=7:

1 = 1/7 + 1/6 + 1/10 + 1/10 + 1/12 + 1/8 + 1/24 + 1/18
+ 1/24 + 1/12 + 1/72 + 1/48 + 1/48 + 1/240 + 1/5040

And for n=11

1 = 1/11 + 1/10 + 1/18 + 1/18 + 1/24 + 1/16 + 1/48 + 1/28 + 1/21
+ 1/56 + 1/28 + 1/168 + 1/30 + 1/24 + 1/36
+ 1/36 + 1/48 + 1/72 + 1/720 + 1/50 + 1/40 + 1/40
+ 1/90 + 1/30 + 1/90 + 1/240 + 1/80 + 1/240
+ 1/3600 + 1/96 + 1/64 + 1/192 + 1/72 + 1/96 + 1/48
+ 1/288 + 1/192 + 1/192 + 1/960 + 1/20160 + 1/324
+ 1/324 + 1/144 + 1/216 + 1/2160 + 1/1152 + 1/288 + 1/576
+ 1/4320 + 1/120960 + 1/3840 + 1/2304
+ 1/5760 + 1/40320 + 1/725760 + 1/39916800

Here is the Sage code that uses Sage's "Partition" function that you can use to print out your own:

Just replace "num" with any positive integer and the code will print out the partition identity. You can also make it faster by removing the part where all the fractions are actually added together, which I put there to make sure it worked.

Do you want to see the result for n=21? I do! Here it is:

1 =
1/21 + 1/20 + 1/38 + 1/38 + 1/54 + 1/36 + 1/108 + 1/68 + 1/51 + 1/136 +
1/68 + 1/408 + 1/80 + 1/64 + 1/96 + 1/96 + 1/128 + 1/192 + 1/1920 + 1/90
+ 1/75 + 1/120 + 1/120 + 1/270 + 1/90 + 1/270 + 1/720 + 1/240 + 1/720 +
1/10800 + 1/98 + 1/84 + 1/140 + 1/140 + 1/168 + 1/112 + 1/336 + 1/252 +
1/336 + 1/168 + 1/1008 + 1/672 + 1/672 + 1/3360 + 1/70560 + 1/104 + 1/91
+ 1/156 + 1/156 + 1/195 + 1/130 + 1/390 + 1/416 + 1/156 + 1/416 + 1/208
+ 1/1248 + 1/468 + 1/468 + 1/312 + 1/468 + 1/4680 + 1/4992 + 1/1248 +
1/2496 + 1/18720 + 1/524160 + 1/108 + 1/96 + 1/168 + 1/168 + 1/216 +
1/144 + 1/432 + 1/240 + 1/180 + 1/480 + 1/240 + 1/1440 + 1/384 + 1/288 +
1/288 + 1/384 + 1/576 + 1/5760 + 1/1944 + 1/432 + 1/1296 + 1/1728 +
1/576 + 1/1728 + 1/25920 + 1/4608 + 1/3456 + 1/11520 + 1/120960 +
1/4354560 + 1/110 + 1/99 + 1/176 + 1/176 + 1/231 + 1/154 + 1/462 + 1/264
+ 1/198 + 1/528 + 1/264 + 1/1584 + 1/550 + 1/220 + 1/330 + 1/330 + 1/440
+ 1/660 + 1/6600 + 1/704 + 1/704 + 1/792 + 1/264 + 1/792 + 1/2112 +
1/704 + 1/2112 + 1/31680 + 1/1782 + 1/1584 + 1/792 + 1/4752 + 1/1584 +
1/1584 + 1/7920 + 1/166320 + 1/42240 + 1/8448 + 1/12672 + 1/63360 +
1/887040 + 1/39916800 + 1/200 + 1/180 + 1/180 + 1/240 + 1/160 + 1/480 +
1/280 + 1/210 + 1/560 + 1/280 + 1/1680 + 1/300 + 1/240 + 1/360 + 1/360 +
1/480 + 1/720 + 1/7200 + 1/500 + 1/400 + 1/400 + 1/900 + 1/300 + 1/900 +
1/2400 + 1/800 + 1/2400 + 1/36000 + 1/960 + 1/640 + 1/1920 + 1/720 +
1/960 + 1/480 + 1/2880 + 1/1920 + 1/1920 + 1/9600 + 1/201600 + 1/3240 +
1/3240 + 1/1440 + 1/2160 + 1/21600 + 1/11520 + 1/2880 + 1/5760 + 1/43200
+ 1/1209600 + 1/38400 + 1/23040 + 1/57600 + 1/403200 + 1/7257600 +
1/399168000 + 1/486 + 1/324 + 1/972 + 1/288 + 1/216 + 1/576 + 1/288 +
1/1728 + 1/315 + 1/252 + 1/378 + 1/378 + 1/504 + 1/756 + 1/7560 + 1/648
+ 1/270 + 1/432 + 1/432 + 1/972 + 1/324 + 1/972 + 1/2592 + 1/864 +
1/2592 + 1/38880 + 1/900 + 1/900 + 1/540 + 1/360 + 1/1080 + 1/810 +
1/1080 + 1/540 + 1/3240 + 1/2160 + 1/2160 + 1/10800 + 1/226800 + 1/3456
+ 1/864 + 1/2304 + 1/1152 + 1/6912 + 1/1296 + 1/1296 + 1/864 + 1/1296 +
1/12960 + 1/13824 + 1/3456 + 1/6912 + 1/51840 + 1/1451520 + 1/17496 +
1/2916 + 1/8748 + 1/7776 + 1/2592 + 1/7776 + 1/116640 + 1/10368 + 1/7776
+ 1/25920 + 1/272160 + 1/9797760 + 1/414720 + 1/69120 + 1/82944 +
1/311040 + 1/2903040 + 1/65318400 + 1/4311014400 + 1/640 + 1/512 + 1/768
+ 1/768 + 1/1024 + 1/1536 + 1/15360 + 1/336 + 1/280 + 1/448 + 1/448 +
1/1008 + 1/336 + 1/1008 + 1/2688 + 1/896 + 1/2688 + 1/40320 + 1/576 +
1/480 + 1/480 + 1/576 + 1/384 + 1/1152 + 1/864 + 1/1152 + 1/576 + 1/3456
+ 1/2304 + 1/2304 + 1/11520 + 1/241920 + 1/1200 + 1/800 + 1/2400 +
1/1280 + 1/480 + 1/1280 + 1/640 + 1/3840 + 1/1440 + 1/1440 + 1/960 +
1/1440 + 1/14400 + 1/15360 + 1/3840 + 1/7680 + 1/57600 + 1/1612800 +
1/3072 + 1/1536 + 1/1536 + 1/2048 + 1/3072 + 1/30720 + 1/5184 + 1/1152 +
1/3456 + 1/4608 + 1/1536 + 1/4608 + 1/69120 + 1/12288 + 1/9216 + 1/30720
+ 1/322560 + 1/11612160 + 1/15552 + 1/10368 + 1/5184 + 1/31104 + 1/6912
+ 1/6912 + 1/34560 + 1/725760 + 1/92160 + 1/18432 + 1/27648 + 1/138240 +
1/1935360 + 1/87091200 + 1/368640 + 1/184320 + 1/368640 + 1/1935360 +
1/23224320 + 1/638668800 + 1/49816166400 + 1/2058 + 1/588 + 1/980 +
1/980 + 1/1176 + 1/784 + 1/2352 + 1/1764 + 1/2352 + 1/1176 + 1/7056 +
1/4704 + 1/4704 + 1/23520 + 1/493920 + 1/1008 + 1/1008 + 1/630 + 1/420 +
1/1260 + 1/1344 + 1/504 + 1/1344 + 1/672 + 1/4032 + 1/1512 + 1/1512 +
1/1008 + 1/1512 + 1/15120 + 1/16128 + 1/4032 + 1/8064 + 1/60480 +
1/1693440 + 1/1400 + 1/1050 + 1/2800 + 1/1400 + 1/8400 + 1/1120 + 1/840
+ 1/840 + 1/1120 + 1/1680 + 1/16800 + 1/5670 + 1/1260 + 1/3780 + 1/5040
+ 1/1680 + 1/5040 + 1/75600 + 1/13440 + 1/10080 + 1/33600 + 1/352800 +
1/12700800 + 1/5376 + 1/5376 + 1/4032 + 1/1344 + 1/4032 + 1/10752 +
1/3584 + 1/10752 + 1/161280 + 1/4536 + 1/4032 + 1/2016 + 1/12096 +
1/4032 + 1/4032 + 1/20160 + 1/423360 + 1/107520 + 1/21504 + 1/32256 +
1/161280 + 1/2257920 + 1/101606400 + 1/27216 + 1/27216 + 1/9072 +
1/13608 + 1/136080 + 1/48384 + 1/12096 + 1/24192 + 1/181440 + 1/5080320
+ 1/80640 + 1/48384 + 1/120960 + 1/846720 + 1/15240960 + 1/838252800 +
1/4515840 + 1/645120 + 1/645120 + 1/1935360 + 1/13547520 + 1/203212800 +
1/6706022400 + 1/610248038400 + 1/3888 + 1/2592 + 1/7776 + 1/1440 +
1/1080 + 1/2880 + 1/1440 + 1/8640 + 1/2304 + 1/1728 + 1/1728 + 1/2304 +
1/3456 + 1/34560 + 1/11664 + 1/2592 + 1/7776 + 1/10368 + 1/3456 +
1/10368 + 1/155520 + 1/27648 + 1/20736 + 1/69120 + 1/725760 + 1/26127360
+ 1/4500 + 1/1200 + 1/1800 + 1/1800 + 1/2400 + 1/3600 + 1/36000 + 1/1920
+ 1/1920 + 1/2160 + 1/720 + 1/2160 + 1/5760 + 1/1920 + 1/5760 + 1/86400
+ 1/4860 + 1/4320 + 1/2160 + 1/12960 + 1/4320 + 1/4320 + 1/21600 +
1/453600 + 1/115200 + 1/23040 + 1/34560 + 1/172800 + 1/2419200 +
1/108864000 + 1/6912 + 1/4608 + 1/13824 + 1/3456 + 1/4608 + 1/2304 +
1/13824 + 1/9216 + 1/9216 + 1/46080 + 1/967680 + 1/7776 + 1/7776 +
1/3456 + 1/5184 + 1/51840 + 1/27648 + 1/6912 + 1/13824 + 1/103680 +
1/2903040 + 1/92160 + 1/55296 + 1/138240 + 1/967680 + 1/17418240 +
1/958003200 + 1/174960 + 1/23328 + 1/69984 + 1/46656 + 1/15552 + 1/46656
+ 1/699840 + 1/41472 + 1/31104 + 1/103680 + 1/1088640 + 1/39191040 +
1/829440 + 1/138240 + 1/165888 + 1/622080 + 1/5806080 + 1/130636800 +
1/8622028800 + 1/3870720 + 1/1658880 + 1/2764800 + 1/11612160 +
1/104509440 + 1/1916006400 + 1/74724249600 + 1/7846046208000 + 1/15000 +
1/6000 + 1/6000 + 1/13500 + 1/4500 + 1/13500 + 1/36000 + 1/12000 +
1/36000 + 1/540000 + 1/4800 + 1/3200 + 1/9600 + 1/3600 + 1/4800 + 1/2400
+ 1/14400 + 1/9600 + 1/9600 + 1/48000 + 1/1008000 + 1/16200 + 1/16200 +
1/7200 + 1/10800 + 1/108000 + 1/57600 + 1/14400 + 1/28800 + 1/216000 +
1/6048000 + 1/192000 + 1/115200 + 1/288000 + 1/2016000 + 1/36288000 +
1/1995840000 + 1/30720 + 1/5760 + 1/15360 + 1/7680 + 1/46080 + 1/5760 +
1/5760 + 1/3840 + 1/5760 + 1/57600 + 1/61440 + 1/15360 + 1/30720 +
1/230400 + 1/6451200 + 1/38880 + 1/6480 + 1/19440 + 1/17280 + 1/5760 +
1/17280 + 1/259200 + 1/23040 + 1/17280 + 1/57600 + 1/604800 + 1/21772800
+ 1/921600 + 1/153600 + 1/184320 + 1/691200 + 1/6451200 + 1/145152000 +
1/9580032000 + 1/145800 + 1/77760 + 1/38880 + 1/233280 + 1/38880 +
1/38880 + 1/194400 + 1/4082400 + 1/345600 + 1/69120 + 1/103680 +
1/518400 + 1/7257600 + 1/326592000 + 1/691200 + 1/345600 + 1/691200 +
1/3628800 + 1/43545600 + 1/1197504000 + 1/93405312000 + 1/51609600 +
1/6451200 + 1/5529600 + 1/13824000 + 1/77414400 + 1/870912000 +
1/19160064000 + 1/871782912000 + 1/104613949440000 + 1/122880 + 1/36864
+ 1/36864 + 1/49152 + 1/73728 + 1/737280 + 1/62208 + 1/13824 + 1/41472 +
1/55296 + 1/18432 + 1/55296 + 1/829440 + 1/147456 + 1/110592 + 1/368640
+ 1/3870720 + 1/139345920 + 1/62208 + 1/41472 + 1/20736 + 1/124416 +
1/27648 + 1/27648 + 1/138240 + 1/2903040 + 1/368640 + 1/73728 + 1/110592
+ 1/552960 + 1/7741440 + 1/348364800 + 1/1474560 + 1/737280 + 1/1474560
+ 1/7741440 + 1/92897280 + 1/2554675200 + 1/199264665600 + 1/233280 +
1/233280 + 1/62208 + 1/93312 + 1/933120 + 1/248832 + 1/62208 + 1/124416
+ 1/933120 + 1/26127360 + 1/276480 + 1/165888 + 1/414720 + 1/2903040 +
1/52254720 + 1/2874009600 + 1/7741440 + 1/1105920 + 1/1105920 +
1/3317760 + 1/23224320 + 1/348364800 + 1/11496038400 + 1/1046139494400 +
1/41287680 + 1/15482880 + 1/22118400 + 1/77414400 + 1/557383680 +
1/7664025600 + 1/199264665600 + 1/10461394944000 + 1/1422749712384000 +
1/11022480 + 1/1049760 + 1/3149280 + 1/1399680 + 1/466560 + 1/1399680 +
1/20995200 + 1/746496 + 1/559872 + 1/1866240 + 1/19595520 + 1/705438720
+ 1/7464960 + 1/1244160 + 1/1492992 + 1/5598720 + 1/52254720 +
1/1175731200 + 1/77598259200 + 1/11612160 + 1/4976640 + 1/8294400 +
1/34836480 + 1/313528320 + 1/5748019200 + 1/224172748800 +
1/23538138624000 + 1/557383680 + 1/61931520 + 1/46448640 + 1/99532800 +
1/464486400 + 1/4180377600 + 1/68976230400 + 1/2092278988800 +
1/125536739328000 + 1/19207121117184000 + 1/3715891200 + 1/1114767360 +
1/1238630400 + 1/3251404800 + 1/16721510400 + 1/153280512000 +
1/2391175987200 + 1/62768369664000 + 1/2845499424768000 +
1/243290200817664000 + 1/51090942171709440000

The first five lines here add up to about 0.486 (!)

Leave a Reply

Your email address will not be published. Required fields are marked *