Math Journey 5:
The Pronic Numbers
“Uncle Bob” Mead
Part 1
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2
4 6
8 10 12
14 16 18 20
22 24 26 28 30
32 34 36 38 40 42
44 46 48 50 52 54 56
…
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The Even Triangle
This journey will focus on the infinite set {2, 6, 12, 20, 30, 42, …}, the pronic numbers, found running down the right edge of the triangular formation of even numbers above. The set has many connections among numbers and geometric models, and I was pleased to stumble on a fresh way to look at them, the Sacks Spiral – it’s the wish for anyone on a journey, right?
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Introduction
Right from the start we have a name problem – or several. I’ve had students in high school and college investigate these numbers, but I never referred to them as pronic numbers. Problem One: The descriptor has no “sizzle,” and as a matter of fact I rate it nerd-plus. Problem Two: Pronic seems to be a misspelling. Problem Three: As many of you know, the subject of mathematics demands precision in its statements, and especially in the definitions of terms. “Pronic” just isn’t precise.
Eric Weisstein of Wolfram explains. “Pronic numbers are also known as oblong (Merzbach and Boyer 1991, p. 50) or heteromecic numbers. However, "pronic" seems to be a misspelling of "promic" (from the Greek promekes, meaning rectangular, oblate, or oblong). However, no less an authority than Euler himself used the term "pronic," so attempting to "correct" it at this late date seems inadvisable.”
[Pronic Number." From MathWorld — A Wolfram Web Resource]
Pronic numbers are the products of two counting numbers that differ by one. There! A concise, precise definition, and the first few examples are: 1 * 2, 2 * 3, 3 * 4, 4 * 5, and so on. I once used the term “oblong” for these, but that term encompasses all non-square rectangles. A 3x5 rectangle is an oblong and 15 is a non-square product, but they are outside of our topic. In our journey, the pronics are nearly square, and if you looked at a drawing of a rectangle that measured 83 by 84 millimeters, it would appear square to you, right? Question for you: Why are the pronic numbers all even?
But what good are they? … the pronics, that is. Here is a little poser, the solution to which employs some of the pronic properties we expose in Part One below.
The Meade School of North Philadelphia had budgeted $702.00 for its field trip to the Franklin and other points downtown. Each student was assessed an equal share of the cost, but just one week before the trip, one student moved away. Fortunately, it was calculated that just one extra dollar per student would cover the full cost. How many students went on the trip?
Part One
Pronics Among the Whole Numbers
“Do you know your times?” is a very common question among the grade school set. Some of the multiplication facts include square and pronic products. In the diagram at right, you can see the square numbers 1, 4, 9, 16, … running down the main diagonal, and running in the same direction just above and just below the squares are the pronic products, one sequence marked with an oblong block.
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Narrowing our focus to just the 5 by 5 portion upper left in the table, the squares max out at 25 and the pronics at 20. Above the squares there is a triangle of entries consisting of the four pronics 2, 6, 12, and 20, topped by 3, 8, and 15, then 4 and 10, and finally 5 in the upper right corner. Due to the symmetry of the chart, those same products make up a triangle below the squares. The counts of products in both triangles is 10 + 10, and that equals 20, our highest pronic in the square. Is that just a coincidence? Let’s look at another triangular formation.
x 1
x x 3 = 1 + 2
x x x 6 = 1 + 2 + 3
x x x x 10 = 1 + 2 + 3 + 4
x x x x x 15 = 1 + 2 + 3 + 4 + 5
… …
Ten is part of a set of triangular numbers. They have the property of being the sum of the counting sequence (1 + 2 + 3 + 4 + …), up to a certain maximum.
The formula that gives triangular sums will lead us to a formula for pronics. The sum of {1, 2, 3, 4} is half of the sum of {1, 2, 3, 4} and {4, 3, 2, 1}. If we pair those up in order and sum we get (1 + 4), (2 + 3), …, or four pairs of 5. So the triangular sum is half of 4*5 or ten. Note that 4*5 is our pronic number 20. Formulas need to work for any size triangle, so let’s get the sum for the set {1, 2, 3, 4, …, n}. We. again sum this set with a reversed duplicate.
1 2 3 4 … n
n n-1 n-2 n-3… 1
We get n pairs each with a sum (n + 1). So our triangular number is n*(n + 1)/2, and more to the point, our pronic formula is n*(n + 1), precisely how they were defined. More on the triangular connection in Part Two of our journey.
Expanding the list. There is no limit to the set of whole numbers, and so the set of pronics is infinite as well. Lots more of them to play with; here are more.
{2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, …}
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Let’s use the function symbol Pr(n) for the nth pronic in the list. The formula becomes Pr(n) = n*(n + 1), and we have found that Pr(4), the fourth pronic, is 4*5 or 20. So Pr(15) = 240 because 240 equals 15*16. We now have two reasons that pronics are all even. They are the double of triangular numbers, and they are the product of an even and an odd number.
Another shape connection. There is a square number exactly half way between consecutive pronics. The mean of 20 and 30 is 25, a square. The mean of Pr(14) and Pr(15) is (210 + 240) /2 or 225, the square of 15. A little algebra shows that this relationship holds for the entire set.
So square numbers are always midway between two consecutive pronics. The shoe on the other foot says that each pronic number lies between two consecutive squares, just as 90 (9*10) lies between the squares 81 and 100, and just a half unit away from their mean of 90.5. At the left we show that this is always the case.
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At the top of this journey we depicted the pronic products on one edge of a triangle of even numbers. We follow with a second triangular model.
The Even Series Triangle
In this model we find the pronics by summing each row. We are doubling up on the model for triangular numbers. Pronics are an accumulation of consecutive even numbers. What is the sum of all even numbers 2 through 100?
2 = 2
6 = 2 + 4
12 = 2 + 4 + 6
20 = 2 + 4 + 6 + 8
30 = 2 + 4 + 6 + 8 + 10
…
A Test for Pronic Numbers
The fact that there is a pronic between consecutive squares allows us to test any number for membership in the set. Is 92 a pronic number? We find its square root to be 9.59 approximately. So 92 lies between the squares of 9 and 10, but 9*10 is not 92. Pick a number at random: is 3,987,554 a pronic number? The square root is 1996.89. We check 1996*1997 which is 3,986,012 and that product is the only pronic in the neighborhood.
Solution to the Meade School Field Trip poser: The full cost $702 is a pronic number, wouldn’t you know. Originally there must have been n students paying x dollars each. But (n – 1) students were charged an extra dollar. Do we need to set up and solve an equation? Nah! To find the factors of 702 that differ by one, simply take the square root: 26.5. So 26 students went on the trip each paying $27.
Part Two
Sums of the Pronic Numbers and Geometric Models
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{2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, …}
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In Part One of our Journey we defined a pronic number as the product of two consecutive counting numbers. You can see a partial list above. We established that there is a square number between successive pronic numbers, for example 16, falling between 12 and 20. The flip side of this is also true: that there is one pronic between successive squares. Pronics and squares play leap frog all the way to infinity. What is the pronic number between 7-squared and 8-squared?
I was very pleased to have made a fresh discovery as I ambled along in this topic. As recently as 2003 software engineer Robert Sacks published a new way to chart the whole numbers. He placed them in a spiral running counterclockwise from zero in the center to each square number on the right, the easterly direction, so that between 1 and 4, the spiral makes space for 2 and 3, and between 4 and 9, the spiral spaces 5, 6, 7, and 8. We’ll see many features of this charting method further along, but for now we merely note that the spiral conveniently locates all the pronic numbers in a westerly direction.
25 + 100*Pr(n) is a square!
Here we have another pronic/square connection: if you tack the digits 25 onto the end of a pronic, it becomes a square number. A simple one to see is 25 appended to the pronic 6, and 625 is the square of 25. Appending 25 onto Pr(15), that’s 25+100*240, we have 24025. You might want to warm up the square root key on your calculator app to see that 24025 is also a square number. A few steps of algebra will take 25+100*n*(n+1) to the expression confirming a square number.
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Pronic Numbers and Triangles
On May 2 this year, our grocery receipt reported that we saved $20.56 buying items that were on sale, and we saved $399.40 so far in 2024. So the almost $400 constituted a running total, an accumulation of the savings in the year to date. We now consider what the numbers look like when we accumulate the pronic numbers, and our path takes us through the triangular numbers again [see more in Part One].
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In one seven-inning softball game, the Teaneck Tigers amassed a passel of runs. The scoreboard showed the runs scored for each inning: 1, 2, 3, 4, 5, 6, 7. That’s right. In each inning they scored one more run than in the previous one. So after two innings they had 3 runs total, and after three innings a total of 6. These are the running totals, and they amount to the triangular sequence 1, 3, 6, 10, 15, 21, and 28 for this game. The sequence is called triangular because it can be modeled as below.
The sequence counts the nodes in our model as more rows are added without limit. We established in Part One that the pronic numbers amount to twice the triangular ones. So the pronic sequence can be modeled by two triangles, one inverted, joined at their apexes. The pronic sequence is extended by growing the model both at the top and at the bottom, as at the right.
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Sums of the Pronic Numbers
What would a running total of the pronic numbers look like? The sequence would begin with 2, then 2 + 6, 2 + 6 + 12, and so on, but these are just doubles of…
Again we start with the triangular sequence and find their sums: 1, 1+3, 1+3+6, 1+3+6+10 and so on. This sequence {1, 4, 10, 20, 35, …} is known as the tetrahedral sequence and, would you believe, we can find it in our multiplication table!
The square numbers are in a diagonal running northwest to southeast, and adding those products gives the sum of squares from one to any limit. The products in diagonals going the other direction, i.e. northeast to southwest, can be summed to give the tetrahedral sequence, e.g., 3, 4, and 3 make 10 in the third of those diagonals; the third tetrahedral number is 10. A tetrahedral shape is a pyramid with a triangular base, so tetrahedral numbers are built from sums of triangular numbers. See below.
We accompany this sequence with a Person X singing “On the first day of Christmas my true love gave to me ….”
The gift types and amounts given to Person X in the Christmas carol are one partridge, two turtle doves, three French hens, four calling birds, five gold rings, on up to 12 something or others. The gifts total much more than 1 + 2 + 3 etc. because all the previous gifts are given again each day. The total gifts given after two days is 1 + 3 or 4: a partridge, two turtle doves, and another blessed partridge. Result: 12 partridges (one each day), 22 doves (2 for each of 11 days), etc., etc.! You’ll find these subtotals marked with a G in the multiplication table. What do the Gs add up to?
The running totals are given by summing the triangular sequence:
1, 1 + 3, 1 + 3 + 6, and so on.
Person Y with Two True Loves
On the first day each of the loves dropped off a partridge. How many gifts is person Y accumulating?
The partial sum sequence is double Person X’s haul: {2, 2+6, 2+6+12, 2+6+12+20, 2+6+12+20+30 …}, which means we are accumulating the pronic numbers.
This sequence begins {2, 8, 20, 40, 70 …}, again all even numbers, and twice
the running totals of the tetrahedral numbers, so our fanciful model is …
... which can be extended at the top and the bottom. Each duplicated tier adds the next pronic, and the running totals top and bottom combined are 2, 8, 20, 40, 70, and so on.
The sum of all the pronics is infinite. The sum of the first k pronics, that is, the sum from 2 through k*(k + 1), is given by
[k*(k + 1)*(k + 2)]/3
For example the third pronic sum, when k = 3, is 3*4*5/3 or 20. This formula doubles the tetrahedral sums sequence.
Sources:
Pronics as figurates
Sack’s spiral
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Next month, in part 3, we reciprocate with pronic reciprocals, and we double down with double pronic numbers. See ya!