SIPPs and Perfect Numbers

If, like me, you find Perfect Numbers fascinating, you are likely to similarly feel awkward at social functions, finding little to say. or respond. I have decided the answer is a SIPP (social interaction preparation plan). This involves deciding, before any such (very rare) social event, on something to say both to introduce a conversation and to respond. I have a few examples which should help you understand.

Obviously, a social interaction is best if it encourages a response. So, I think a question format is ideal. In order to encourage involvement it would also be good if there was an element of disclosure involved and possible some intimacy. Something like “I think it best to deal with blackheads by squeezing them straight away. Don’t you?” might work well. But too intimate is not good. Even if the circumstances were favourable I would not advise anything along the lines of “Would you mind telling me if my garlic mouth wash is curing my halitosis?”.

Having questions ready is not sufficient for a SIPP. You also need to prepare suitable ripostes otherwise the conversation dies and you are back to square one. You might, if you were a masculine male in a matey conversation, be faced with an approach like “I find Angela Merkel very sexy. Don’t you?” in which case having an answer which carries the conversation forward such as “Yes, I think it is her jackets that get me going. What about you?” Of course, if you are a female or a homosexual-looking male, the subject of the conversation would probably be Herman van Rompuy. (I have just discovered from Wikipedia that his middle name is Achille! and that he is younger than me. Two very surprising things – for me anyway.) An appropriate ready answer might refer to his hungry features. I would like to hear if others can help me develop my SIPP.

English: Herman Van Rompuy, President of the E...

English: Herman Van Rompuy, President of the European Council, at the press conference about the European Union during the 37th G8 summit in Deauville, France. (Photo credit: Wikipedia)

I recently read about perfect numbers. I had a vague memory of them but needed reminding about detail. A perfect number is a positive integer that is the sum of its proper divisors ( Thus, 6 is the first perfect number because the proper divisors of 6 are 1,2 and 3 and 1+2+3=6. I would have thought that this was quite a simple requirement to meet and there would be lots of perfect numbers. But no. The next one is 28 (1,2,4 and 7) then 496, then 8128. All these were known to the ancient Greeks and were considered by some to have mystical significance, eg 28, the number of days in the lunar cycle; an idea taken up and used by some early Christians. For example, St Augustine in The City of God wrote “Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect.” Though how he could possibly know is unexplained.

After the fourth, perfect numbers become quite large:

  • Fifth 33,550,336
  • Sixth 8,589,869,056
  • Seventh 137,438,691,328

If you want to see further examples look at where the date of discovery is also given.

You may have noticed that all these numbers are even. Nobody has managed to prove that there are no odd perfect numbers but if there are then the number is greater than 10 to the power 1500! One thing you probably have not noticed and I think quite remarkable is that, except for the first number each is the sum of the cubes of consecutive odd numbers 28= 1*1*1+3*3*3 and 496= 1*1*1+3*3*3+5*5*5+7*7*7. I won’t do the others. But if you would like some homework…

Another spooky thing, but one that follows from their definition is that the sum of the reciprocals of the divisors adds up to 2 exactly. Thus, 1/1+1/2+1/3+1/6 = 2 for 6 the first perfect number.

The perfect numbers also have a notable structure when expressed in binary form as illustrated below

Base 10 Base 2
6 110
28 11100
496 111110000
8128 1111111000000
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