The sum of two cubes in two different ways

This is a continuation of an entry last March on Ramanujan and the sums of two cubes. If the title makes no sense, start there.

This expands the list of solutions, and below it, I reveal my Diophantine approach to spreadsheets. I should note here that one of my most successful fraud investigations (the precise details of the client remain commercial-in-confidence) only worked because I used a spreadsheet to divine the cunning way a wages swindle was being run.

All I needed was a stapler and a pie chart, but to find out how I managed to get the data and wring out the answer, you need to ply me with beer.

If you make it a Guinness, I will also throw in a practical demonstration of isostasy: look behind the harp in this shot, and see if you can spot it.

Anyhow, here are my solutions:

-9728: 16³ + -24³; -12³ + -20³

-7657: -20³ + 7³; -17³ + -14³

-6832: -19³ + 3³; -18³ + -10³

-5859: -18³ + -3^3; -19³ + 10³

-5824: -18³ + 2³; -16³ + -12³

-5256: -17³ + -7³; -20³ + 14³

-5103:-15³ + -12³; -18³ + 9³

-4104: -15³ + -9³; -16³ + -2³; -18³ + 12³ TRIPLE!

-3367: -16³ + 9³; -15³ + 2³

-3087: -14³ + -7³; -20³ + 17³

-2457: -18³ + 15³; -12³ + -9³

-1736: -12³ + -2³; -18³ + 16³

-1729: -9³ + -10³; -1³ + -12³

-1512: -10³ + -8³; -12³ + 6³

-1216: -12³ + 8³; -10³ + -6³

-1027: -19³ + 18³; -10³ + -3³

-999: -12³ + 9³; -10³ + 1³

-728: -12³ + 10³; -9³ + 1³; -8³ + -6³ TRIPLE!

-721: -16³ + 15³; -9³ + 2³

-513: -9³ + 6³; -8³ + -1³

-217: -6³ + -1³; -9³ + 8³

-189: -6³ + 3³; -5³ + -4

-152: -5³ + -3³; -6³ + 4³

-91: -4³ + -3³; -6³ + 5³

91: 4³ + 3³; 6³ + -5³

152: 5³ + 3³; 6³ + -4³

189: 6³ + -3³; 5³ + 4

217: 6³ + 1³; 9³ + -8³

513: 9³ + -6³; 8³ + 1³

721: 16³ + -15³; 9³ + -2³

728: 12³ + -10³; 9³ + -1³; 8³ + 6³ TRIPLE!

999: 12³ + -9³; 10³ + -1³

1027: 19³ + -18³; 10³ + 3³

1216: 12³ + -8³; 10³ + 6³

1512: 10³ + 8³; 12³ + -6³

1729: 9³ + 10³; 1³ + 12³

1736: 12³ + 2³; 18³ + -16³

2457: 18³ + -15³; 12³ + 9³

3087: 14³ + 7³; 20³ + -17³

3367: 16³ + -9³; 15³ + -2³

4104: 15³ + 9³; 16³ + 2³; 18³ + -12³ TRIPLE

5103:15³ + 12³; 18³ + -9³

5256: 17³ + 7³; 20³ + -14³

5824: 18³ + -2³; 16³ + 12³

5859: 18³ + 3^3; 19³ + -10³

6832: 19³ + -3³; 18³ + 10³

7657: 20³ + -7³; 17³ + 14³

9728: -16³ + 24³; 12³ + 20³

Here is the base spreadsheet that spat out all the sums of two cubes where the source numbers were -20 to 20:

This is a bit kludgy, but once row 2 was done, I could use FILL-DOWN.

Next, I copied all of the sums of two cubes into a document, sorted them into order, and pasted them into column A of a fresh spreadsheet in column A. Column B had in B2  =IF(A1=A2, "hit","").  

Next, I deleted the non-hit rows, and knew what to look for: here's a sample of the target list:

Now I was ready to search the first spreadsheet, and this was where I copied Diophantus. I might have come up with a routine that did it for me, but it was faster and less brain-damaging just to hack away.

I have been known to sneer at geography a colouring-in, but as I am colour blind, I think I'm allowed to do that. Anyhow, I used colouring-in to pin down my pairs:

All that remained to do now was to check the cells and enter the source values, but as I was doing this manually, I set a check in column E, where E1 read =C1*C1*C1+D1*D1*D1, followed by COPY-DOWN.

For more on 1729, see https://oldblockwriter.blogspot.com/2015/05/adventures-with-1729.html or

https://oldblockwriter.blogspot.com/2019/02/of-cubes-and-smart-lazy.html or

https://oldblockwriter.blogspot.com/2022/03/was-ramanujan-wrong-or-wrongly-reported.html

My book of STEAM activities for bright young minds, Playwiths, has a number of labour-saving ideas involving spreadsheets, and about 1729 and Ramanujan, but now I need to add some extra stuff, based on what I have written here.

There was a time when hacking was an honourable activity, when the Kalashnikov rifle was described as a hack, and my fraud-busting spreadsheet was most emphatically a hack. We cannot turn the linguistic clock back to save hack, but we can at least try to make recreational computing respectable again!

 

Spittlebugs

 When I was a teenager, the act of spitting was a golly or gollying, and just lately, North Head has been taking a bombardment of gollies, going on appearances. Bushes all over the headland are garlanded like this, with festoons of foam, as though some grubby louts have just passed through.

The 'gollies' are made by small bugs called spittlebugs, and when I say "bugs", I am using entomologist-speak, so my bugs are Hemiptera, insects with piercing mouthparts, like the cicadas, aphids and bird-of-paradise flies (which I will get around to describing one day).

These insects feed on plant sap, and make a sticky foam that seems to keep predators away. Working as a volunteer in a sanctuary, I was asked last Tuesday to prune back some trees that get a bit exuberant. We don't like doing this, but it reduces the fire hazard, so we do it with care. One Monotoca was home to many spittlebugs, and as one of the pockets in my work trousers always has a supply of jars, I snipped away a few of them, and by the time I got home, they had emerged from their foam to see what was going on. So out came the macro lens and the microscope.

First, here we have a juvenile and an adult (some people call them froghoppers): they were both in a Petri dish, which is the curve you can see.

But how big are they? One reason for using a Petri dish is that you cam slip a millimetre scale beneath it: if you let insects loose on a scale, they always seem to run away.

Now let's look at the juveniles:

There's more to say about these beasties, but as I'm flat out (as usual), here's a useful link. I will be playing with these for a few weeks, and this week, I was showing some to Stage 1 kids at Manly Vale, so here are some better shots:

If you want more stuff like this, you could do worse than look into my The Nature of North Head.  The price can be as low as $3 for non-dead-tree versions, $20 for monochrome, but the one in bookshops is colour and dead tree at $50.

It's been a while

 I am travelling right now, but service will be resumed shortly.