the difference of the products cannot be divisible by the prime, and consequently all products of the base by numbers less than the prime must, on division by the prime, give different remainders. But the remainder after division by a number is of course less than that number. Hence the products of the base by the different numbers less than the prime will after division by the prime give those same numbers as remainders, in some different order. Thus, suppose the prime is 5, and the base 6. Then the products of 6 by the numbers, 1, 2, 3, 4, less than 5, are
6 = 5 · 1 + 1, 12 = 5 · 2 + 2, 18 = 5 · 3 + 3, 24 = 5 · 4 + 4.
But if 7 had been the base, the products would have been 7, 14, 21, 28, which leave as remainders after division by 5, 2, 4, 1, 3, the same numbers in a different order. Consequently, the product of all the products of the base by the different numbers less than the prime (which for the sake of brevity may be called the product of products) […]
[Reasoning Exercises]
| Winter 1887 | Houghton Library |
[Number Series]
The following rows of numbers are called Fermat’s series:
0 1 3 7 15 31 63 127 255 511 1023 2047 etc.
2 3 5 9 17 33 65 129 257 513 1025 2049 etc.
Find out the rules of the succession of numbers in these two series.
The numbers in the following row are called the phyllotactic numbers. The series is also called Fibonacci’s series, because first studied in the XIIth century by the mathematician Leonardo of Pisa, called Fibonacci.
0 1 1 2 3 5 8 13 21 34 55 89 144 etc.
Find out the rule of succession of these numbers.
Do the same for the series
2 1 3 4 7 11 18 29 47 76 123 199 322 etc.
The following series are called Pell’s series:
0 1 2 5 12 29 70 169 408 985 2378 etc.
2 2 6 14 34 82 198 478 1154 2786 6726 etc.
Find the rule of succession for these.
0 1 1 0 −1 −1 0 1 1 0 −1 −1 etc.
Find the rule of succession here.
What is the product of two corresponding numbers in Fermat’s series?
Compare the square of any number in any series with the products of those which precede and follow it. What rules can you find?
If two numbers one over another in any pair of series are both prime, what is true of the number of their place in the series? (N.B. The places are to be numbered beginning with 0.)
What relation can you find between the phyllotactic number whose place in the series is expressed by the sum of any two numbers, say m and n, and the numbers in the mth and nth places in that series and in the series underneath it?
Every third number in the series of Fibonacci is even, every 4th number is divisible by 3. What about every 5th number, every 6th etc.? Is there anything analogous in any of the other series?
Compare the greatest common divisor of two phyllotactic numbers with the greatest common divisor of their places in the series.
Give the rule or plan of each arrangement.
What is the relation between arrangements B and F? What between C and D? What is so related to E?
Show that as B is to A, so is D to B, E backwards to D, C to E backwards, F to C, and A backwards to F.
[Relational Graphs]
Required to make some graphs in which every spot is just like every other and has connections exactly like those of every other spot. See how many different forms of graphs you can make under these conditions, with 3, with 4, and so on up to 12.
Make some graphs in which the spots are of two colors and in each color of two shapes and the lines are of two kinds and every line runs from a spot of one color to a spot of another color and the lines are not barbed. See how many different kinds of graphs you can draw under these conditions beginning with the simplest.
Fig. 1
Fig. 2
Fig. 3
Let the black spots • denote points and the circles ˚ lines, and let the connections signify that points lie on lines. Thus
Problem: to draw straight lines so that the state of things represented in Fig. 3 shall be carried out.
Fig. 4
Fig. 5
Fig. 6
Figs. 4 and 6 are substantially the same, but Fig. 5 is different.
Fig. 7. Draw two triangles, having their vertices on three lines that meet in one point. Then the corresponding sides, produced if necessary, meet in 3 points lying on a 10th line. Show how this graph represents the relations either of the points or lines.
Fig. 8. The dots stand for the edges of a cube. Show that the junctions may signify either that two edges meet or that they do not lie in one plane.
Fig. 9
| ⊙ |
An instant of time. Joined by barbed line |
| • |
A person or firm. |
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