The discussion revolves around finding all possible integer solutions (p, q) for the equation involving a nested square root repeated 1964 times: sqrt(p + sqrt(p + sqrt(...(p + sqrt(p))...))) = q. Participants explore various mathematical approaches, including squaring both sides and testing specific values, while considering the implications of the finite repetition of the square root.
Discussion Character
Exploratory
Technical explanation
Mathematical reasoning
Debate/contested
Main Points Raised
One participant suggests squaring both sides as a potential method to simplify the equation.
Another participant proposes that the integer solutions might be of the form (x^2 - x, x) for non-negative integers x, but later revises this to conclude that (0, 0) is the only solution.
It is noted that if q is an integer, then p must be a perfect square, leading to the conclusion that k must equal 0 for the equation to hold.
Some participants express uncertainty about whether other solutions exist beyond (0, 0), with one stating they were unable to prove the non-existence of additional solutions.
Another participant introduces a sequence approach, suggesting that for certain values of p, the sequence stabilizes, yielding solutions (0, 0) and (2, 2).
There is a discussion about the implications of the infinite case versus the finite case of the square root, with some participants clarifying their confusion regarding the problem's conditions.
One participant explores a generalized version of the problem, seeking integer triplets (p, q, r) with similar nested square roots, but only identifies (2, 2, 2) and (0, 0, 0) as solutions.
Areas of Agreement / Disagreement
There is no consensus on the existence of solutions beyond (0, 0), with multiple competing views and uncertainty expressed regarding the completeness of the solution set. Some participants agree on the solutions found, while others remain skeptical about the possibility of additional solutions.
Contextual Notes
Participants note that the problem's specific condition of having the square root repeated exactly 1964 times may limit the applicability of general solutions derived from infinite cases. There are also discussions about the implications of negative values for p and the necessity for p to be non-negative for q to remain an integer.
#1
K Sengupta
113
0
Determine all possible integer solutions (p,q) of the equation:
sqrt(p+ sqrt(p+ sqrt(...(p + sqrt(p))...))) = q
The "sqrt" symbol in the above relationship is repeated 1964 times.
I don't see an immediate way to do it, but by testing in Haskell I can tell you that the answer will probably be the set of all (x^2-x, x) for non-negative integer x except for the pair (0, 1) which is not present. Unless my results are misleading due to rounding error.
Edit: actually now I'm thinking the only solution is (0, 0). If x is not an integer and p is an integer, then sqrt(p + x) is not an integer. Therefore by induction if any of the terms within any of the square root signs are not integers, q is not an integer. So if q is an integer, then
sqrt(p + sqrt(p)) is integer. But that requires that p is a perfect square, so p = k^2 for some k. Then p + sqrt(p) = k + k^2 = k(k+1) is also a perfect square. The only non-negative integer that satisfies that is k = 0.
Last edited:
#4
K Sengupta
113
0
On Today 03:13 PM; Office_Shredder wrote:
PHP:
Try squaring both sides
I am a little confused.
If the relationship :
sqrt(p+ sqrt(p+ sqrt(...(p + sqrt(p))...))) = q ...(*)
was inclusive of the repetition of the "sqrt" symbol an infinite number of times; one would readily have obtained p = q^2 - q; giving an infinite number of solutions to the problem for an integer p.
However, it may be noted that in terms of the problem, the "sqrt" symbol is repeated precisely 1964 times in (*), and consequently I am not very sure that the parametric relationship p = q^2 - q; would satisfy the conditions of the problem.
Your solution corresponds to the equation:
sqrt(p+ sqrt(p+ sqrt(...(p + sqrt(p+p))))) = q; where “sqrt” symbol is repeated 1964 times……………………..(#)
rather than the original problem.
We consider the sequence S_1(p) = S_1(p) = sqrt(p + p); and
S_(i+1)(p) = 2+ sqrt(S_i(p)) for i = 1,2,…..,1963.
For p=0 and 2, we readily observe that S_(i+1)(p) = S_i(p); for all non-negative integers i.
So, S_(1964)(p) = q; yields (p,q) = (0,0) and (2,2) as two of the integer solutions to (#).
I thank you for the new problem and, if possible I will try to obtain other values for (p,q) or try to prove the non-existence of any other integer solutions to (#).
However, it may be noted that in terms of the problem, the "sqrt" symbol is repeated precisely 1964 times in (*), and consequently I am not very sure that the parametric relationship p = q^2 - q; would satisfy the conditions of the problem.
You're right, the ... in the sqrt confused me. Sorry about that
If the relationship :
sqrt(p+ sqrt(p+ sqrt(...(p + sqrt(p))...))) = q ...(*)
was inclusive of the repetition of the "sqrt" symbol an infinite number of times; one would readily have obtained p = q^2 - q; giving an infinite number of solutions to the problem for an integer p.
Except for the case 0 = 1^2 - 1--(0, 1) does not satisfy that infinite equation.
#12
K Sengupta
113
0
On Today at 12:04 AM; 0rthodontist wrote:
PHP:
Except for the case 0 = 1^2 - 1--(0, 1) does not satisfy that infinite equation.
True. I stand corrected.
The parametric solution p=q^2-q, for the infinite equation:
q = sqrt(p+(sqrt(p+…..sqrt(p+sqrt(p))))); where both p and q are integers and; the “sqrt” symbol is repeated 1964 times………….(i)
suggests that p must be non-negative.
Otherwise, for any given negative p, sqrt(p) is never a real number, which is a contradiction.
Since from (i) , q>= sqrt(p); for non-negative p; it follows that q must be non-negative.
Accordingly, we only need to consider only the non-negative roots of the equation:
q^2 – q = p.
Now, q^2-q –p =0, gives:
q = (1+/- sqrt(4p+1))/2
For p>=1; q = (1- sqrt(4p+1))/2, is negative which is a contradiction.
So, q = (1+sqrt(4p+1))/2; for all p>=1 …….(ii)
For p=0, we obtain q^2 = q; so that, q = 0,1.
A direct check with equation (i) would reveal that:
p=0 yields q=0 and (p,q) = (0,1) is a contradiction.
In relationship (ii); the minimum value of p >=1 for which sqrt(4p+1) is an integer occurs at p=2; giving:
q = (1+3)/2=2 as the minimum value of q whenever p>=1.
Subsequently; the required integer solution for the infinite equation (i) would be:
(p,q) = (0, 0) and:
p= q^2-q; whenever q>=2.
arildno‘s problem requires one to determine all possible integer solutions to the undernoted equation:
sqrt( p+sqrt(p+(……+sqrt(p+sqrt(2p))…)).= q; where both p and q are integers and the “sqrt” symbol is repeated 1964 times…………..(#)
We observe that:
S_1(p) = sqrt(2p);
S_2(p) = sqrt(p+sqrt(2p));
S_3(p) = sqrt( p+ sqrt(p+sqrt(2p))
S_(i+1)(p) = sqrt( p+ sqrt(S_i(p))); for i = 2,3,…,1963.
Now, S_1(p) is an integer only when p =2* k^2, for some integer k.
Accordingly, S_2(p) = sqrt(2k(k+1)).
Clearly, for q to correspond to an integer, it follows that S_2(p) must be an integer; so that:
2k(k+1) = (S_2(p))^2
Solving the underlying Pell’s equation for k<300; we obtain:
(k,p,S_1(p), S_2(p), S_3(p))
= (1,2,2,2,2); (8,128, 16, 12, sqrt(140)); (49, 4802, 98, 70, sqrt(4872));
(288, 165888, 576, 408, sqrt(166296))
Now,
S_1(p) = S_i(p) = 0, whenever p=0; for i=2,3,….,1964
S_1(p) = S_i(p) = 2, whenever p=2; for i = 2,3,…,1964
Since S_3(p) does not correspond to an integer for p< 2*300^2 = 180,000; we can safely conclude that the equation (#) admits of the solution:
(p,q) = (0,0); (2,2) whenever p<180,000.