Puzzle 1113 A pair of Diophantine equations.

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Puzzle 1113 A pair of Diophantine equations.

Paul Cleary sent the following nice puzzle:

Given the two equations

x^2 + 1200*y = n^2

y^2 + 1200*x = m^2

x > y, and both positive integers.

When and x = 157 and y = 143 both n and m are the primes 443 and 457 respectively.

Q1. Find more values for x and y, where x > y that make both n and m primes.

Q2. Can you tell how many distinct prime-solutions in total are there for this pair of equations?

During the week Nov 26-Dic 2, 2022, contributions came from Giorgos Kalogeropoulos, Paul Cleary, Adam Stinchcombe, Gennady Gusev, Ken Wilke, Emmanuel Vantieghem, Oscar Volpatti

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Giorgos wrote:

These two families of curves (where the second one is rotated 90 degrees) intersect infinitely often.

This is why I believe we can find infinitely many lattice points {x,y} although they are pretty sparse.

Here are the 41 solutions that I found with n and m primes:

{ x , y , n , m }
{157,143,443,457}
{161,139,439,461}

{167,133,433,467}

{179,121,421,479}

{191,109,409,491} x,y are also primes

{199,101,401,499} x,y are also primes

{203,97,397,503}

{221,79,379,521}

{241,59,359,541} x,y are also primes

{247,53,353,547}

{263,37,13,313} x,y are also primes

{269,31,331,569} x,y are also primes

{287,13,313,587}

{293,7,307,593} x,y are also primes

{407,169,607,719}

{1469,523,1669,1427}

{3689,2743,4111,3457}

{4301,2611,4651,3461}

{5269,2821,5581,3779}

{5819,487,5869,2687}

{6649,5773,7151,6427}

{10421,139,10429,3539}

{15181,2233,15269,4817}

{19409,19109,19991,19709}

{27071,18547,27479,19403}

{53983,16583,54167,18433}

{96889,3553,96911,11353}

{108521,1447,108529,11503}

{160867,17699,160933,22501}

{170051,14173,170101,20123}

{171127,98783,171473,99817}

{212471,2833,212479,16217}

{261299,871,261301,17729}

{1085171,90433,1085221,97367}

{1724659,235709,1724741,240059}

{1906159,737093,1906391,738643}

{1926173,815467,1926427,816883}

{2156089,79057,2156111,94007}

{2991173,518479,2991277,521929}

{25257869,2609983,25257931,2615783}

{113476571,1513021,113476579,1557371}

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Paul wrote:

This is my method and solutions to this puzzle.

Given

x^2 + 1200y = n^2

y^2 + 1200x = m^2

since x and y are positive we can write

x^2 + 1200y = (x + a)^2 and

y^2 + 1200x = (y + b)^2.

Where a and b are positive integers, after expanding we find the square terms cancel leaving just linear simultaneous equations thus.

1200y = 2 a x + a^2

1200x = 2 b y + b^2

Solving for x and y we get the following

x = (2*a^2 b + 1200 b^2)/(1440000 - 4 a b) and

y = (2*a b^2 + 1200 a^2)/(1440000 - 4 a b).

and since a and b are positive, the numerators in each fraction must be positive, so to have a positive denominator

a*b < 360000. with Judicial selection of a and b, all solutions (3535 in total) can be found in under a minute. However of those 3535 solutions only 48 with x>y>0 and both m and n prime.

Here are all my solutions.

...

Regarding the solutions sent by Giorgos, Paul noted:

All the solutions submitted are correct, though there are 7 solutions missing I’m, surprised Giorgos missed this one. {313, 13, 337, 613}. Then there are these after his last solution.

{210972899, 703243, 210972901, 864707}

{213602921, 2848039, 213602929, 2892689}

{271181099, 903937, 271181101, 1068887}

{995465099, 3318217, 995465101, 3493583}

{7287489299, 24291631, 7287489301, 24470969}

{24246030899, 80820103, 24246030901, 80999903}.

Regarding the claim of infinite solutions, or lattice points for {x,y} I don’t think there are, there are only some 3535 solutions in total, If there are I would be interested in any with an x, y value greater than, x = 2429946090299 and y = 8099820301.

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Adam wrote:

Up through max(n,m)=the 1807 th prime, I found 24 sets of n,m,x,y:

[443, 457, 157, 143], [439, 461, 161, 139], [433, 467, 167, 133], [421, 479, 179, 121], [409, 491, 191, 109], [401, 499, 199, 101], [397, 503, 203, 97], [379, 521, 221, 79], [359, 541, 241, 59], [353, 547, 247, 53], [337, 563, 263, 37], [331, 569, 269, 31], [313, 587, 287, 13], [307, 593, 293, 7], [337, 613, 313, 13], [607, 719, 407, 169], [1427, 1669, 523, 1469], [3457, 4111, 2743, 3689], [3461, 4651, 2611, 4301], [3779, 5581, 2821, 5269], [2687, 5869, 487, 5819], [6427, 7151, 5773, 6649], [3539, 10429, 139, 10421], [4817, 15269, 2233, 15181]

For seven of these, all x,y,n,m are prime.

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Gennady wrote:

Q1. There were found 34 solutions for x,y < 10^6.
x y n m
------ ------ ------ ------
157 143 443 457
161 139 439 461
167 133 433 467
179 121 421 479
191 109 409 491
199 101 401 499
203 97 397 503
221 79 379 521
241 59 359 541
247 53 353 547
263 37 337 563
269 31 331 569
287 13 313 587
293 7 307 593
313 13 337 613
407 169 607 719
1469 523 1669 1427
3689 2743 4111 3457
4301 2611 4651 3461
5269 2821 5581 3779
5819 487 5869 2687
6649 5773 7151 6427
10421 139 10429 3539
15181 2233 15269 4817
19409 19109 19991 19709
27071 18547 27479 19403
53983 16583 54167 18433
96889 3553 96911 11353
108521 1447 108529 11503
160867 17699 160933 22501
170051 14173 170101 20123
171127 98783 171473 99817
212471 2833 212479 16217
261299 871 261301 17729

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Ken wrote:

Attached is my comment on Puzzle 1113. It finds other members of a family of solutions based upon the example solution given. Here may be additional soluti9ons, but I did not see any decent route to them.

Solution:
In this solution, we consider only those solutions which are the same type as the example
given by the proposer. Consider the following equations in which x, y, m and n and all
positive integers with x >y, m>n and both m and n are primes.
x^2 + 1200*y = n^2 (1)
y^2 + 1200*x = m^2 (2)
Subtracting equation (2) from equation (1) yields
x^2–y^2 + 1200*(y –x) = n^2 – m^2 which can be rewritten as
(x–y)[(x+y)-1200] = (n-m)(m+n) (3)
According to the sample solution provided, we have x-y = m-n. Thus since m+n > x+y,
x+ y > 1200 is impossible so equation (3) becomes
1200 = (m+n)+(x+y) (4).
Noting that in the given solution, m+n =457 + 443 = 900 and x+y = 300 we search a table
of primes for pairs of primes m and n such that m+n =900. That search reveals 14 pairs of
primes, including the pair in the given solution,(m,n) =(457,443), (461,439), (467,433),
(479, 421), (491,409),(499, 401), (503,397), (521,379), (541,359), (547,353), (563,337),
(569,331), (587,313)and (593, 307). The corresponding pairs of x and y are respectively
(x,y) can be computed easily since m-n = x-y and m+n =900.
Then (m,n,n,x,y)=(457,443,157,143),(461,439,161,139),(467,433,161,139),
(479,421,179,121),((491,409,191,109),(499,401,199,101),(503,397,203,97),
(521,379,221,79),(541,359,241,59),(547,353,247,53),(563,337,263,37),(569,331,269,31),
(587,313,287,13)and (593,307,293,7).

Of particular interest are (m,n,x,y) =
(491,409,191,109),(499,401,199,101),(503,397,203,97),(541,359,241,59),
(563,337,263,37),(569,331,269,31),(587,313,287,13) and (593,307,293,7)
in which each of m, n, x and y are prime.

***

Emmanuel wrote:

Q1

Here are all the solution I found for x, y, m, n :
157, 143, 457, 443
161, 139, 461, 439
167, 133, 467, 433
179, 121, 479, 421
191, 109, 491, 409 *
199, 101, 499, 401 *
203, 97, 503, 397
221, 79, 521, 379
241, 59, 541, 359 *
247, 53, 547, 353
263, 37, 563, 337 *
269, 31, 569, 331 *
287, 13, 587, 313
293, 7, 593, 307 *
313, 13, 613, 337 *
407, 169, 719, 607
1469, 523, 1427, 1669
3689, 2743, 3457, 4111
4301, 2611, 3461, 4651
5269, 2821, 3779, 5581
5819, 487, 2687, 5869
6649, 5773, 6427, 7151
10421, 139, 3539, 10429
15181, 2233, 4817, 15269
19409, 19109, 19709, 19991
27071, 18547, 19403, 27479
53983, 16583, 18433, 54167
96889, 3553, 11353, 96911
108521, 1447, 11503, 108529
160867, 17699, 22501, 160933
170051, 14173, 20123, 170101
171127, 98783, 99817, 171473
212471, 2833, 16217, 212479
261299, 871, 17729, 261301
1085171, 90433, 97367, 1085221
1724659, 235709, 240059, 1724741
1906159, 737093, 738643, 1906391
1926173, 815467, 816883, 1926427
2156089, 79057, 94007, 2156111
2991173, 518479, 521929, 2991277
25257869, 2609983, 2615783, 25257931
113476571, 1513021, 1557371, 113476579
210972899, 703243, 864707, 210972901
213602921, 2848039, 2892689, 213602929
271181099, 903937, 1068887, 271181101
(The seven solutions marked with * consist of four primes).

Q2.

I think there might be infinitely many solutions.

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Oscar wrote:

There are only 48 pairs (x,y) with 0<y<x such that n and m are both prime.
Let's assume that positive integers x,y,n,m are a solution.
n^2 = x^2+1200*y > x^2,
m^2 = y^2+1200*x > y^2.
n is greater than x and has the same parity of x, so n = x+2*a for some positive integer a;
m is greater than y and has the same parity of y, so m = y+2*b for some positive integer b.

By substituting and simplifying, we obtain a system of equations which is linear in x and y:
300*y = a*x + a^2,
300*x = b*y + b^2.
If a*b = 90000 the system is inconsistent, otherwise it has a unique solution:
x = b*(300*b+a^2)/(90000-a*b),
y = a*(300*a+b^2)/(90000-a*b).

I'll prove that the upper bound a*b < 90000 must hold.

Consider the first equation:
300*y = a*x + a^2.

Solve for x in terms of y:
x = (300/a)*y - a.
Drop the term "- a", obtaining a strict inequality:
x < (300/a)*y.
Now consider the second equation:
(y+2*b)^2 = y^2 + 1200*x.
Substitute "x" with "(300/a)*y", obtaining another strict inequality:
(y+2*b)^2 < y^2 + (360000/a)*y.

Add the positive term "(180000/a)^2" to the right side only, so that inequality still holds:
(y+2*b)^2 < y^2 + (360000/a)*y + (180000/a)^2.

Now the right side is the square of a positive quantity too:
(y+2*b)^2 < (y+180000/a)^2.

Take the square root of both sides, then cancel y out:
2*b < 180000/a.

Multiply both sides by "a/2":
a*b < 90000.
QED.

Therefore, only 1040555 pairs (a,b) are permitted.
For each permitted pair of positive integers (a,b), we obtain a unique pair of positive rationals (x,y).
Inequality y<x also holds if and only if a<b; this further reduces the search space to 520128 pairs (a,b).
Only 3535 permitted pairs (a,b) actually produce an integer pair (x,y).

Only 48 such pairs (a,b) also produce a prime pair (n,m).

x y n m
157 143 443 457
161 139 439 461
167 133 433 467
179 121 421 479
191 109 409 491
199 101 401 499
203 97 397 503
221 79 379 521
241 59 359 541
247 53 353 547
263 37 337 563
269 31 331 569
287 13 313 587
293 7 307 593
313 13 337 613
407 169 607 719
1469 523 1669 1427
3689 2743 4111 3457
4301 2611 4651 3461
5269 2821 5581 3779
5819 487 5869 2687
6649 5773 7151 6427
10421 139 10429 3539
15181 2233 15269 4817
19409 19109 19991 19709
27071 18547 27479 19403
53983 16583 54167 18433
96889 3553 96911 11353
108521 1447 108529 11503
160867 17699 160933 22501
170051 14173 170101 20123
171127 98783 171473 99817
212471 2833 212479 16217
261299 871 261301 17729
1085171 90433 1085221 97367
1724659 235709 1724741 240059
1906159 737093 1906391 738643
1926173 815467 1926427 816883
2156089 79057 2156111 94007
2991173 518479 2991277 521929
25257869 2609983 25257931 2615783
113476571 1513021 113476579 1557371
210972899 703243 210972901 864707
213602921 2848039 213602929 2892689
271181099 903937 271181101 1068887
995465099 3318217 995465101 3493583
7287489299 24291631 7287489301 24470969
24246030899 80820103 24246030901 80999903

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