Test Your Intuition (16): Euclid’s Number Theory Theorems

Euclid’s

Euclid’s book IX on number theory contains 36 propositions.

The 36th proposition is:

Proposition 36.If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect.

It asserts that if 2^n-1 is a prime number then 2^{n-1}\cdot (2^n-1) is a perfect number. (A number m is perfect of it is equal to the sum of its proper divisors.)

This is certainly a remarkable achievement of ancient Greek mathematics. Other Propositions of the same book would be less impressive for us:

Proposition 23.If as many odd numbers as we please are added together, and their multitude is odd, then the sum is also odd.

Proposition 24.If an even number is subtracted from an even number, then the remainder is even.

Proposition 25.If an odd number is subtracted from an even number, then the remainder is odd.

Proposition 26.If an odd number is subtracted from an odd number, then the remainder is even.

Proposition 27.If an even number is subtracted from an odd number, then the remainder is odd.

Proposition 28.If an odd number is multiplied by an even number, then the product is even.

Proposition 29.If an odd number is multiplied by an odd number, then the product is odd.

Test your intuition: What is the reason that deep mathematical results are stated by Euclid along with trivial results.

Test Your Intuition (15): Which Experiment is More Convincing

Consider the following  two scenarios

(1) An experiment tests the effect of a new medicine on people which have a certain illness. The conclusion of the experiment is that for 5% of the people tested the medication led to improvement while for 95% it had no effect. (The experiment followed all rules: it had a control test it was double blind etc. …)

A statistical analysis concluded that the results are statistically significant, where the required statistical significance level is 1%. This roughly means that the probability p_1  that such an effect happend by chance (under the “null hypothesis”) is less or equal 0.01. (This probability is called the p-value. Suppose that p_1= 0.008.)

(2) An experiment tests the effect of a new medicine on people which have a certain illness. The conclusion of the experiment is that for 30% of the people tested the medication led to improvement while for 70% it had no effect. (The experiment followed all rules: it had a control test it was double blind etc. …)

A statistical analysis concluded that the results are statistically significant, where the required statistical significance level is 1%. (Again, this roughly means that the probability p_2  that such an effect happend by chance (under the “null hypothesis”) is less or equal 0.01. And again suppose that p_2= 0.008.)

Test your intuition: In which of these two scenarios it is more likely that the effect of the medication is real.

You can assume that the experiments are identical in all other terms that may effect your answer. E.g., the theoretical explanation for the effect of the medicine. Note that our assumption p_1=p_2 is likely to imply that the sample size for the first experiment is larger.