Wednesday, September 21

13-digit 2-balanced factorization integers

When I introduced my k-balanced factorizations last month, I noted that there were over 13000 of the 2-balanced variety. My effort to chart them all is proceeding slowly, with a couple of months yet to go before I reach 10^12. In the meantime I decided to have a try at the ones greater than 10^12. This can be accomplished by brute-forcing all possible ways of creating a valid factorization using seven or fewer digits. I generated no solutions for one to four digits. For five digits, one gets the 4 solutions of the known 1-balanced factorizations (that started it all):

        26487 = 3^5 * 109
        28651 = 7 * 4093
        61054 = 2 * 7^3 * 89
        65821 = 7 * 9403

For six digits, there are again no solutions. For seven digits, there are 4 solutions:

1495476527089 = 83^2 * 601^3
3392164558027 = 7^9 * 84061
8789650571264 = 2^31 * 4093
9418623046875 = 3^9 * 5^10 * 7^2

These then are (necessarily) the largest 2-balanced factorization integers! I'm going to run the program again for eight digits. This will generate all 12-digit 2-balanced factorization integers and may do so (I'm hoping) in less time than my exhaustive forward search. In addition, there's a chance that it may find a k > 2 solution, since nothing in the reverse search precludes it.

Monday, September 19

Eastern milk

The only snakes I have ever seen here in Toronto are garters. So when I came across this deceased (note the ants) juvenile by the curb in the road, I had to look up what kind it was. I have since been told that an adult eastern milk snake was spotted by a homeowner across the street from this scene. That person's home is right above the new retaining wall on the Humber river. So this snake's habitat has been disrupted by the construction, resulting in its being noticed where it might otherwise never have ventured. Hopefully it survives.

Thursday, September 15

Curb your enthusiasm

The sealant in the expansion joint between a sidewalk-proper and its curb degrades over time, allowing a few hardy species to gain a foothold. I assume there's a root system that manages to find its way beneath the concrete. Gotta love these plants!

Sunday, August 28

A factorization balancing act

A couple of weeks ago, Claudio Meller presented 26487 and 65821 as examples of the property of having one each of the base-ten digits when combined with the digits of their respective factorizations. Surprisingly, he missed two:

    26487 = 3^5 * 109
    28651 = 7 * 4093
    61054 = 2 * 7^3 * 89
    65821 = 7 * 9403

I wondered how this might be turned into a sequence. Base-ten k-balanced factorization integers: The combined digits of an integer and its factorization primes and exponents contain exactly k copies of each of the ten digits. So,

 45849660 = 2^2 * 3 * 5 * 19 * 37 * 1087
 84568740 = 2^2 * 3 * 5 * 67 * 109 * 193
104086845 = 3^2 * 5 * 19 * 23 * 67 * 79
106978404 = 2^2 * 3 * 13 * 685759

and so on. For any given k, k-balanced integers are necessarily finite. For k=2, there are a little over 13000. Is the largest of these greater than the smallest 3-balanced integer?

It's not too difficult to generate very large terms:


is an example of a 13-balanced integer. Can you come up with a larger one?

Saturday, August 6

My 100th Leyland prime find

Last October I found my first previously unknown Leyland (probable) prime. Today I found my 100th. The graph shows (in order of discovery) the number of decimal digits of those 100 primes, ranging from 43633 to 61184. The finds have had the unintended (but certainly not unpleasant) consequence of pushing me up a list of probable prime contributors! My search for Leyland primes is of course in aid of my Leyland prime indexing effort which has reached #1137. There'll be a significant jump in about three weeks when I finish rounding up the few remaining unknowns with decimal digit lengths of between 54334 and 55390.

Saturday, July 30

Point of entry

We have a totally fenced yard at the back and sides of the house. To be more accurate, a monster shed on the adjacent property at the back of the yard takes the place of a fence. The front fencing on the (gravel) driveway side accommodates a gate that can be swivelled to allow vehicles into the yard. It is this gate that is being compromised.

For a number of years now some creature has been creating a summer underpass here to allow entry into (and presumably exit from) the yard. I have a shovel nearby to refill the hole and have come to place some small concrete chunks on the yard side of the gate to give the visitor a nasty surprise half-way through its dig. Sometimes a new underpass is generated to one side of the concrete but more generally the creature gives up.

I had always thought the offender was a skunk but some weeks ago Bodie and I were surprised — in broad daylight — by a rat running through the yard, only to disappear (after it was surprised by us) under that back shed!

Friday, July 29


Not too far from our home, this car-crash situation stopped Weston Road traffic both ways for hours. Not seen in the photo are two additional cars to the left, one of which seems to have sustained significant damage. It is difficult for me to imagine a scenario that would flip a car just so but there it is. Notice a woman's shoe sitting on the undercarriage. I haven't found a hint of this event in the media!

Thursday, July 28

Farmers' market

The long-running Weston farmers' market has moved a little closer to my home this year because of impending construction at the old (parking lot) site. I rarely go but decided to accompany Catherine (green hat, far left, below) back on the 16th. It's much smaller than the old site and — in terms of local produce or decent deals — a little disappointing.

Monday, June 20

Cubic Lock

These are the four pieces of Cubic Lock by Goh Pit Khiam as realized in exotic woods by Brian Menold. A union of the two pieces on the left (top, the key made up of 10 cubies; bottom, 9 cubies) is inserted into a union of the two pieces on the right (top, 23 cubies; bottom, 19 cubies) and by means of a strategic to-and-fro of the key, the pieces are shifted into place. It's not obvious that the finished 4*4*4 cube has an internal void until one counts and adds those cubies. A good show-off puzzle as one is unlikely to forget the assembly once one has put it together a few times.

Wednesday, June 8

The wall

I went to Raymore Park (on the other side of the Humber river) last Friday to see what progress had been made on the erosion-control retaining-wall they were putting in place (on my side of the river). Contrary to the intelligence in my previous entry on this, I can now see that the wall will not be so much "on top of the now-in-the-river foundation" as I had supposed but (rather) much higher and more-closely hugging the slope bedrock — which is actually being exposed for a more stable conglomeration. And that storm drain interruption will end up being a barrier to my ever walking along the full length of the wall. At any rate (depending on the wall-top width), it may be too high to be safe. The bits of white floating through the air (more evident in the second photo) is tree fluff.