7
7 7 days in a week, and therefore associated with 14 and with 28 days in a lunar month. The 4th prime number, and the first of the form 6n + I. The start of an arithmetical progression of six primes: 7, 37, 67, 97, 127, 157. 7 and II are the first pair of consecutive primes different by 4. The 3rd Mersenne number, 7 = 2 3 - I, and the second Mersenne prime, leading to the second perfect number. The first number that is not the sum of at most 3 squares. The sequence of such numbers continues, 15 23 28 31 39 47 55 60 ... 7 = 3! + l. n! + I is prime for n = 1,2,3, 11,27,37,41,73,77,116, 154, 320, 340, 399,427, and no other values below 546. Brocard's problem. When is n! + I a square? The only known solutions are n = 4, 5 and 7: 7! + I = 5041 = 7P. The Fermat quotient 2P - 1 p
is a square only when pis 3, or 7. Lame proved in 1840 that Fermat's equation, x 7 + y7 = Z7 has no solutions in integers. If a, b are the shorter sides of a Pythagorean triangle, then 7 divides one of a, b, a - b or a + b. Because 72 falls short of 50 by only I, 7 was called by the Greeks, the rational diagonal of a square of side 5. All sufficiently large numbers are the sum of 7 positive cubes. To test if a number is divisiblc by 7: multiply the Icft-hand digit by 3 and add the next digit. Repeat as often as necessary. If the final answer is divisible by 7, so is the original number. Alternatively, start by multiplying the right-hand digit by 5 and adding the adjacent digit. Repeat as before. 7 numbers are sufficient to colour any map on a torus. Surprisingly, this was known before the 4-colour conjecture was solved for plane maps. At least 7 rectangles are required if a rectangle is to be divided into smaller rectangles no one of which will fit inside another. The smallest rectangle that can be tiled 'incomparably' is 13 by 22. * At least 7 rectangles are also required to divide a rectangle into smaller rectangles of different shape but equal area. • A. C. C Yao and E. M. Reingold, Journal (1 Re"ealiOlwl Malhel/ialiCl, vol M
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