TY - JOUR
T1 - Solutions of f (n) = f (n + k) and s (n) = s (n + k)
AU - Ford, Kevin
N1 - Publisher Copyright:
© The Author(s) 2020. Published by Oxford University Press. All rights reserved.
PY - 2022/3/1
Y1 - 2022/3/1
N2 - We show that for some even k 3570 and all k with 442720643463713815200|k, the equation f(n) = f(n+k) has infinitely many solutions n, where f is Euler's totient function. We also show that for a positive proportion of all k, the equation s(n) = s(n + k) has infinitelymany solutions n. The proofs rely on recent progress on the prime k-tuples conjecture by Zhang, Maynard, Tao, and PolyMath.
AB - We show that for some even k 3570 and all k with 442720643463713815200|k, the equation f(n) = f(n+k) has infinitely many solutions n, where f is Euler's totient function. We also show that for a positive proportion of all k, the equation s(n) = s(n + k) has infinitelymany solutions n. The proofs rely on recent progress on the prime k-tuples conjecture by Zhang, Maynard, Tao, and PolyMath.
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U2 - 10.1093/imrn/rnaa218
DO - 10.1093/imrn/rnaa218
M3 - Article
AN - SCOPUS:85110912883
SN - 1073-7928
VL - 2022
SP - 3561
EP - 3570
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 5
ER -